On the Foundations of Arithmetic in Euclid

David Pierce

Draft of January , 

Mathematics Department Mimar Sinan Fine Arts University Istanbul http://mat.msgsu.edu.tr/~dpierce/

Summary

The geometry presented in the thirteen books of Euclid’s Elements is founded on five postulates; but the arithmetic in Books vii, viii, and ix makes no explicit use of postulates. Nonetheless, Euclid proves rigorously the commutativity of multiplication in any ordered ring whose positive elements are well ordered: such is one conclusion of this monograph, worked out in the third and last chapter. The various contents of the chapters and their sections might be summarized as follows. . The study of Euclid is an instance of doing history. As such, the study can both benefit from, and illustrate, the philosophy of history developed by R. G. Collingwood in several of his books. .. Studying the Elements is like studying an ancient building, such as the Hagia Sophia in Istanbul: it is a kind of archeology. Archeology is in turn a kind of history. Historical inference actually resembles mathematical inference; but the way to understand this is by ac- tually doing history and mathematics—which we do by studying Euclid. .. The present study of Euclid is motivated by questions such as: Can students today learn number theory from Euclid? Does Euclid cor- rectly prove such results as the commutativity of multiplication, and “Euclid’s Lemma”? .. We cannot decide whether a particular statement or argument of Euclid is correct or incorrect without first understanding what it means. The meaning is not always obvious. .. In studying Euclid, we have to re-enact his thoughts, as best we can. .. In reading Euclid (or any philosopher), we cannot just say that he is wrong (if we think he is), without also offering a correction that (in our best judgment) he can agree with. . Mathematicians used to learn mathematics from Euclid. Since this

  Summary no longer happens, we may be better able now to understand Euclid prop- erly. Nonetheless, some students do still learn mathematics from Euclid. Among those students are undergraduates in my own mathematics de- partment in Istanbul: reading with them brings out certain features of Euclid. .. Dedekind did not learn his construction of the real numbers from Euclid. Unlike some of his contemporaries, he understood Euclid well enough to see that what he was doing was different. .. Looking back at Euclid’s Greek (for the sake of translating this into Turkish) brings out some misleading features of the standard En- glish translation by Heath. Heath aids the reader by typographical means; but this may cause us to think wrongly that Euclid’s propo- sitions are like modern theorems. Euclid may have established a pattern for modern mathematical exposition; but this does not mean he is obliged to follow it. .. Contrary to modern mathematical practice, Euclid’s equality is not sameness or identity. Thinking about what equality really means, one can see that Euclid’s Proposition i., the “Side Angle Side” the- orem of triangle congruence, is a real theorem. .. According to Hilbert, Euclid’s Fourth Postulate, the equality of all right angles, is really a theorem. Examining Hilbert’s proof of this theorem shows how different is his way of thinking of geometry from Euclid’s. . We turn to Book vii of Euclid’s Elements and to the specific inves- tigation of questions raised earlier. .. The definition of unity may well be a late addition to this book. .. By definition, if a number is of a second number the same part or parts that a third number is of a fourth, then the four numbers are proportional. Whatever else it means, a proportion is not an equation of ratios, but an identification of them. .. Euclid’s numbers are finite nonempty sets. Euclid distinguishes be- tween dividing numbers into parts and measuring numbers by num- bers. It turns out that for infinite sets, being divisible into two equipollent subsets is logically stronger than being measurable by a two-element set. 

.. If a number is parts of a number, this does not mean that the one number is a fraction of the other. .. When the numbers in Euclid’s diagrams appear as line segments, it is useful to think of these as lyre strings. The ancient musical treatise called Sectio Canonis suggests a way of thinking about numerical ratios that is useful for understanding the Elements. .. Lying behind the notion of a ratio is the alternating subtraction or anthyphaeresis used in the Euclidean Algorithm. .. Proposition , that the less number is either part or parts of the greater, should be understood as an explanation of what being the same parts means in the definition of proportion: it means that application of the Euclidean Algorithm requires the same pattern of alternating subtractions in either case. .. With this understanding, the proof of the commutativity of multi- plication, Proposition , unfolds logically. .. So does the proof of “Euclid’s Lemma,” Proposition . .. The last three propositions of Book vii may be a late addition. In any case, they are written by somebody who is starting to think of mathematics as symbol-manipulation. Contents

Summary 

 Philosophy of History  . Archeology ......  . Questions ......  . Evidence......  . Re-enactment......  . Science...... 

 Euclid in History  . DedekindandHilbert ......  . FromacourseofEuclid ......  . Equality......  . Rightangles...... 

 Euclid’s Foundations of Arithmetic  . Unity ......  . Proportion ......  . Sets ......  . Parts......  . Music ......  . The Euclidean Algorithm ......  . PartorParts ......  . Multiplication......  . Euclid’sLemma......  . Symbolic mathematics ...... 

 List of Figures

. Proposition i. ......  . Proposition i. ......  . Proposition i.......  . Proposition i.......  . Proposition i.asEucliddoesit ......  . Proposition i. ......  . Proposition i. ......  . Postulate ...... 

. Multiplication of points ......  . Multiplication of straight lines ......  . Lyrestrings ......  . Proposition vii. ......  . Proposition vii. ......  . Proposition i. ...... 

  Philosophy of History

. Archeology

We are going to investigate foundational aspects of Euclid’s arithmetic, as presented in Books vii, viii, and ix of the thirteen books of the Elements. Our investigation might be called archeology, though it will require no actual digging. We want to know something about the Elements, and in particular its theory of numbers; however, we have no first-hand tes- timony about what Euclid was doing, or trying to do. We just have the Elements itself. As for the earlier tradition that Euclid came out of, we have only traces of it. We have later works about Euclid, especially Proclus’s Commentary on the First Book of Euclid’s Elements. We shall make some use of this commentary, but it was written centuries after the Elements. Euclid himself does not provide testimony about what he is doing; he just does it. If proper history requires such testimony, then we cannot make an historical study of Euclid’s arithmetic. We can still make an archeological study. We can read the Elements itself, for evidence of what Euclid was trying to do there. In the same way, one might read the Constantinopolitan Church of St Sophia, for evidence of the aims of its master builders, Anthemius of Tralles and Isidore of Miletus: for, the basilica itself can still be visited in Istanbul. Concerning this basilica, called the Hagia Sophia or Ayasofya, we do also have some written testimony. Procopius saw the church constructed in the sixth century, at the command of Emperor Justinian, after an

A Wikipedia article requires testimony in a broader sense. Because of the rule of No Original Research, a Wikipedia article about a book must be based on what other people have said about the book.

 . Archeology  earlier church had been destroyed in the so-called Nika Insurrection. Jus- tinian spared no expense to build a new church, says Procopius,

so finely shaped, that if anyone had enquired of the Christians before the burning if it would be their wish that the church should be destroyed and one like this should take its place, shewing them some sort of model of the building we now see, it seems to me they would have prayed that they might see their church destroyed forthwith, in order that the building might be converted to its present form. [, I.i., p. ] We might infer that Anthemius and Isidore did actually show Justinian a model of the basilica that they planned to build. Like this model, or like the old church that the new Hagia Sophia would replace, some kind of mathematics came before the Elements as we have it now; but we have only hints (as in Plato’s Meno) of what this was like. About the construction of the Hagia Sophia itself, from Procopius we know

it was not with money alone that the Emperor built it, but also with labour of the mind and with the other powers of the soul, as I shall straightway show. One of the arches which I just now mentioned (lôri the master-builders call them), the one which stands towards the east, had already been built up from either side, but it had not yet been wholly completed in the middle, and was still waiting. And the piers (pessoi), above which the structure was being built, unable to carry the mass which bore down upon them, somehow or other suddenly began to crack, and they seemed on the point of collapsing. So Anthemius and Isidorus, terrified at what had happened, carried the matter to the Emperor, having come to have no hope in their technical skill. And straightway the Emperor, impelled by I know not what, but I suppose by God (for he is not himself a master-builder), commanded them to carry the curve of this arch to its final completion. “For when it rests upon itself,” he said, “it will no longer need the props (pessoi) beneath it.” And if this story were without witness, I am well aware that it would have seemed a piece of flattery and altogether incredible; but since there were available many witnesses of what then took place, we need not hesitate to proceed to the remainder of the story. So the artisans carried out his instructions, and the whole arch then hung secure, sealing by experiment the truth of his idea. Thus, then, was this arch completed. [, I.i.–, pp. –],   Philosophy of History

We have no such contemporary account of the Elements. Nobody who knew Euclid can tell us, for example, of Euclid’s discovery of the Fifth or Parallel Postulate, as being needed to prove such results as the equality of an exterior angle of a triangle to the two opposite interior angles. Nobody can testify that, since the Parallel Postulate is phrased in terms of right angles, Euclid figured he needed another postulate, according to which all right angles were equal to one another, and thus arose the Fourth Postulate. We have no testimony that Euclid’s thoughts proceeded in this way; we can only infer it (or something like it) from the edifice of the Elements as it has come down to us. By some accounts then, there can be no properly historical study of the Elements, but only a “prehistorical” or archeological study. However, I am going to agree with the philosopher and historian R. G. Collingwood (–) that archeology is history. It is hard to say that we know more about the construction of the Hagia Sophia than of the Elements, simply because we have Procopius’s fanciful story about how the Emperor saved a partially erected arch from collapse. That story might after all be true; but it can hardly be credited without independent reason for thinking it plausible. I shall make use of Collingwood’s ideas about history, because I know them and find them relevant. Now, the Hagia Sophia is relevant as being, like the Elements, one of the great structures of the ancient world. After the Turkish conquest of Constantinople, nine centuries after the construc- tion of the Hagia Sophia, this edifice and especially its great dome became a model for Ottoman Imperial mosques []. Euclid’s Elements became a model for mathematics, even as it is done today. As for Collingwood, though some of his books have remained in print or been brought back into print, he is little known today. After speculating on why this is, Simon Blackburn concludes,

A lucky life, then, rather than an unlucky one, is the explanation for Collingwood’s unattractive features—unless, as Aristotle thought, we

One might classify the Hagia Sophia as medieval rather than ancient. The edifice at least symbolizes the ancient mathematical world, in that the master builder Isidore also compiled texts of Archimedes with the commentaries of Eutocius, and he is mentioned in the extant manuscripts [, p. , n. ]. . Archeology 

cannot even call men lucky when they die, since we can be harmed after our death. I do not know whether Aristotle was right, but if he was wrong, then in most respects the neglect of Collingwood’s thought may be our tragedy rather than his. []

In Blackburn’s judgment, “Collingwood was the greatest British philoso- pher of history of the twentieth century” []. In her own assessment of Collingwood, Mary Beard argues,

it is surely crucial that he was a product of the old Oxford ‘Greats’ (that is, classics) course, which focused the last two and a half years of a student’s work on the parallel study of ancient history on the one hand, and ancient and modern philosophy on the other. Most students were much better at one side than the other . . . In the context of Greats, Collingwood was not a maverick with two incompatible inter- ests. Given the educational aims of the course, he was a rare success, even if something of a quirky overachiever; his combination of interests was exactly what Greats was designed to promote. []

Collingwood’s combination of interests does not seem to have included mathematics especially. Nonetheless, he was aware of it and talked about it, as will be seen below. His sense of what it means to do history will help us as we bring our own interest in mathematics to the reading of Euclid. In directing actual archeological excavations, Collingwood found him- self “experimenting in a laboratory of knowledge,” as he reports in An Autobiography of  [, p. ]. The study of Euclid can likewise serve as a laboratory of ideas about history. Collingwood likens history to mathematics in The Idea of History, posthumously edited and first pub- lished in :

One hears it said that history is ‘not an exact science’. The meaning of this I take to be that no historical argument ever proves its conclu- sion with that compulsive force which is characteristic of exact science. Historical inference, the saying seems to mean, is never compulsive, it is at best permissive; or, as people sometimes rather ambiguously say, it never leads to certainty, only to probability. Many historians of the present writer’s generation, brought up at a time when this proverb   Philosophy of History

was accepted by the general opinion of intelligent persons (I say noth- ing of the few who were a generation ahead of their time), must be able to recollect their excitement on first discovering that it was wholly untrue, and that they were actually holding in their hands an historical argument which left nothing to caprice, and admitted of no alternative conclusion, but proved its point as conclusively as a demonstration in mathematics. [, pp. –] These words were originally intended for The Principles of History [, p. ], whose extant manuscripts were rediscovered in , having been thought discarded after being (severely) edited to form part of The Idea of History. I do not know whether historians generally agree with Colling- wood on the subject of historical inference. According to one archeolo- gist’s explanation of his field,

Since nobody knows what happened in the past (even in the recent historical past), there will never be an end to archaeological research. Theories will come and go, and new evidence or discoveries will alter the accepted fiction that constitutes the orthodox view of the past and which becomes established through general repetition and widespread acceptance. As Max Planck wrote, ‘A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents die and a new generation grows up that is familiar with it.’ Archaeology is a perpetual search, never really a finding; it is an eternal journey, with no true arrival. Everything is tentative, nothing is final. [, p. ] We shall return later (in §., p. ) to Paul Bahn’s skepticism here. His reference to “accepted fiction” calls to my mind Gore Vidal’s  Introduction to his  historical novel Julian []. Vidal quotes one historian on another as being “The best in the field. Of course he makes most of it up, like the rest of us.” Speaking for himself, Vidal says,

Why write historical fiction instead of history? Because, when dealing with periods so long ago, one is going to make a lot of it up anyway, as Finley blithely admitted. Also, without the historical imagination even conventional history is worthless. Finally, there is the excitement when a pattern starts to emerge. . Archeology 

An emerging pattern in history is the opportunity for the flights of fancy that Vidal takes in novels like Julian and Creation. Vidal does however admit a difference between fiction and history, as when he says, as every dullard knows, the historical novel is neither history nor a novel . . . I personally don’t care for historical novels as such (I’m obliged to read history in order to write stories set in the past). In any case, our present topic is not history as such, but Euclid. The editors of The Principles of History lament that, when declaring the compulsive force of historical inference, Collingwood does not give “at this point an example of historians actually reasoning to certain conclusions” [, p. xxix]. But one should be able to supply one’s own examples. Euclid will be our example, and in a footnote, the editors supply theirs. They report that, in a  article, Collingwood said, “it is more certain than ever that the forts [on Hadrian’s Wall] were built before the stone Wall.” This conclusion is now held to be incorrect. Collingwood’s supposed historical mistake does not disprove his ideas of historical inference, any more than mathematical mistakes dispel our belief that mathematical correctness is possible. I suppose every mathe- matician has had the pleasure and excitement of discovering a theorem, only to find later that it was not a theorem after all. We may then give up mathematics, or we may simply return to it with more care. Again, if Collingwood does not give examples of historical inference in The Principles of History, it is because he expects readers to come up with their own. He has already said, Like every science, history is autonomous. The historian has the right, and is under an obligation, to make up his own mind by the methods proper to his own science as to the correct solution of every problem that arises for him in the pursuit of that science . . . There is no need for me to offer the reader any proof of this statement. If he knows anything of historical work, he already knows of his own experience that it is true. If he does not already know that it is true, he does not know enough about history to read this essay with any profit, and the best thing he can do is to stop here and now. [, p. ] [, pp. –] Actually, the best thing the reader could do would be to do some history— which is just what we are going to do with Euclid.   Philosophy of History

. Questions

In his own experiments in the historical or archeological “laboratory of knowledge,” Collingwood was

at first asking myself a quite vague question, such as: ‘was there a Fla- vian occupation on this site?’ then dividing that question into various heads and putting the first in some such form as this: ‘are these Fla- vian sherds and coins mere strays, or were they deposited in the period to which they belong?’ and then considering all the possible ways in which light could be thrown on this new question, and putting them into practice one by one, until at last I could say, ‘There was a Flavian occupation; an earth and timber fort of such and such plan was built here in the year a b and abandoned for such and such reasons in the ± year x y.’ Experience soon taught me that under these laboratory ± conditions one found out nothing at all except in answer to a question; and not a vague question either, but a definite one. [, p. ]

Concerning Euclid’s number-theory, my own vague initial question is, “Can Books vii–ix of the Elements serve as a text for undergraduates today, as Book i can?” Book i is indeed a text for students in my own mathematics department, as will be discussed in §. (p. ). Euclid does prove several famous results that those students should also learn: • the efficacy of the “Euclidean Algorithm” for finding greatest com- mon divisors (vii., ); • “Euclid’s Lemma,” that a prime divisor of the product of two num- bers divides one of the numbers (vii.); • that the set of prime numbers is unbounded (ix.); n k • that if the sum Pk=0 2 of consecutive powers of two (starting from one) is a prime p, then its product p 2n with the last term in the sum is a perfect number (ix.). · Towards formulating a more precise question than whether Euclid is suit- able as a textbook, I note a certain unsuitability in the textbooks of today.

The result is called Euclid’s Lemma by Burton [, p. ] and by Wikipedia; Hardy & Wright [, p. , Thm ] call it Euclid’s First Theorem (the Second being, “The number of primes is infinite”). . Questions 

There is a common foundational error of assuming that recursive defini- tions of number-theoretic functions are justified by induction alone. Thus if we say 1! = 1 and (k +1)! = k! (k + 1), it may be thought that () we have defined n! when n = 1, and· () if we have defined n! when n = k, then we have defined it when n = k + 1; therefore, “by induction,” we have defined n! for all natural numbers n. Actually cannot define it by induction alone. We have made a recursive definition, relying on k + 1 to uniquely determine k, and requiring 1 not to be of the form k + 1. Clar- ification of this point leads to insight, as I have argued elsewhere []. It so happens that induction is enough to justify the recursive definitions of addition and multiplication, as Landau shows tacitly in The Founda- tions of Analysis []. Therefore congruence of integers with respect to a particular modulus respects addition and multiplication, since induction is valid on integers considered with respect to a modulus. One can then use induction to prove associativity and commutativity of addition and multiplication, as well as distributivity of the latter over the former. The proofs of these results can be exercises for students today. Euclid himself establishes commutativity of multiplication in Proposition vii. of the Elements. A more definite question for the present investigation is then, “Does Euclid give a valid proof of the commutativity of multiplication?” Another question is whether Euclid gives a valid proof of “Euclid’s Lemma.” The question is investigated by Pengelley and Richman [] and I need then by Mazur []; the former also review earlier, albeit still modern, Pengelley’s investigations. Such investigations can be inhibited by the modern notion more recent paper of a fraction. In the Elements, there are two definitions of proportion, to be quoted in full in §. (p. ): there is a clear definition of proportion of magnitudes at the head of Book v, and an unclear definition of proportion of numbers at the head of Book vii. By the latter definition, is A to B as C to D, just in case the fractions A/B and C/D are equal? To answer this, we should ask what question Euclid was trying to answer in his writing.

The phrase “with respect to the modulus m” is, in Gauss’s Latin, secundum modulum m [, §]. For reasons unknown to me, the English translation [] sticks to Latin, in the shorter form “modulo m.”   Philosophy of History

I shall propose that Euclid’s question is, “Can we simplify the anthy- phaeretic definition of proportion, that is, the definition based on the method of finding greatest common divisors [the Euclidean Algorithm]?” I think this proposal is at least hinted at by David Fowler in The Mathe- matics of Plato’s Academy [], which is all about anthyphaeresis, though Really? it does not say too much about Euclid’s arithmetic as such. I am aware of other writing on Euclid and his theory of numbers; there may be much more of which I am not aware. As an apology for just going ahead anyway, and saying what I have to say, I cite Robert Pirsig, who coins a word in his philosophical novel Lila [, ch. , pp. –] and defines it by an analogy:

Philosophology is to philosophy as musicology is to music, or as art history and art appreciation are to art, or as literary criticism is to creative writing.

One might add two more terms to the analogy, namely history of math- ematics and mathematics itself. And yet, as has already been suggested, you cannot properly study the history of mathematics without already being something of a mathematician. The point for now is that, according to Pirsig, philosophologists put

a philosophological cart before the philosophical horse. Philosophol- ogists not only start by putting the cart first; they usually forget the horse entirely. They say first you should read what all the great philoso- phers of history have said and then you should decide what you want to say. The catch here is that by the time you’ve read what all the great philosophers of history have said you’ll be at least two hundred years old.

Some questions about Euclid have arisen in the course of my own mathe- matical life, and here I take them up. In the next chapter, I shall consider the question of how Richard Dedekind and David Hilbert (and my own students) read Euclid. One could raise the same question about every earlier mathematician, since just about all of them must have read Eu- clid. But one has only so much time. One should also have something one is looking for: this was said by Collingwood above, and it is said by Pirsig, or more precisely by his persona, called Phaedrus: . Evidence 

Phaedrus, in contrast, sometimes forgot the cart but was fascinated by the horse. He thought the best way to examine the contents of various philosophological carts is first to figure out what you believe and then to see what great philosophers agree with you. There will always be a few somewhere. These will be much more interesting to read since you can cheer what they say and boo their enemies, and when you see how their enemies attack them you can kibitz a little and take a real interest in whether they were right or wrong.

It is unfortunate that Pirsig thinks of intellectual life as an arena for attack and defense; but apparently there are reasons for this.

. Evidence

We can read Euclid as if he were a contemporary mathematician. Then we may think that we understand him if we have translated his mathe- matics into our own terms. This is the approach that E. C. Zeeman takes and defends in an article (originally a talk) on Euclid:

In our discussions we found ourselves following the traditional opposing roles of historian and mathematician. The historian thinks extrinsi- cally in terms of the written evidence and adheres strictly to that data, whereas the mathematician thinks intrinsically in terms of the mathe- matics itself, which he freely rewrites in his own notation in order to better understand it and to speculate on what might have been passing through the mind of the ancient mathematician, without bothering to check the rest of the data. [, p. ]

The person given the role of historian here is the late David Fowler, mentioned above; but he was a mathematician as well. Zeeman seems to suggest that the mathematician has an advantage over the historian, because an understanding of mathematics can take a researcher places where a lack of evidence prevents the historian from going. Though we have seen (or at least seen it argued by Collingwood) that mathematics is similar to history, the two disciplines are still different. Collingwood suggests this himself in The Principles of History [, p. ]. Mathematics and history are sciences; but   Philosophy of History

anything that is a science at all must be more than merely a science, it must be a science of some special kind. A body of knowledge is never merely organized, it is always organized in some particular way. There are sciences of observation, like meteorology; there are sciences of experiment, like chemistry. There is mathematics, not named as such by Collingwood here, but described as being

organized not by observing events at all, but by making certain as- sumptions and proceeding with the utmost exactitude to argue out their consequences. History is not like this. In meteorology and chemistry, the aim is “to detect the constant or recurring features in all events of a certain kind.” In being different from this, history does resemble mathematics, but only to a certain point:

It is true that in history, as in exact science, the normal process of thought is inferential; that is to say, it begins by asserting this or that, and goes on to ask what it proves. But the starting-points are of very different kinds. In exact science they are assumptions, and the tradi- tional way of expressing them is in sentences beginning with a word of command prescribing that a certain assumption be made: ‘Let ABC be a triangle, and let AB = AC.’ In history they are not assumptions, they are facts, and facts coming under the historian’s observation, such as, that on the page open before him there is printed what purports to be a charter by which a certain king grants certain lands to a cer- tain monastery. The conclusions, too, are of different kinds. In exact science, they are conclusions about things which have no special habi- tation in space or time: if they are anywhere, they are everywhere, and if they are at any time they are at all times. In history, they are conclusions about events, each having a place and date of its own. Collingwood appears to have the view of mathematics described by Tim- othy Gowers in Mathematics: A Very Short Introduction [, pp. –]. Gowers recognises it as a twentieth-century view. It is a view that has its place in history:

[T]he steps of a mathematical argument can be broken down into smaller and therefore more clearly valid substeps. These steps can . Evidence 

then be broken down into subsubsteps, and so on. A fact of fundamen- tal importance to mathematics is that this process eventually comes to an end . . . What I have just said in the last paragraph is far from obvious: in fact it was one of the great discoveries of the early th century, largely due to Frege, Russell, and Whitehead. This discovery has had a profound impact on mathematics, because it means that any dispute about the validity of a mathematical proof can always be resolved. In the th century, by contrast, there were genuine disagreements about matters of mathematical substance. For example, Georg Cantor, the father of modern set theory, invented arguments that relied on the idea that one infinite set can be ‘bigger’ than another. These arguments are accepted now, but caused great suspicion at the time.

Cantor’s arguments are accepted now, and one might say this is because they can be worked out in a formal system based on the Zermelo–Fraenkel theory of sets. Does Gowers mean to suggest that any dispute about the validity of a proof in Euclid can now be resolved? I do not know; but I myself would say that today’s mathematics does not automatically give us a good criterion for assessing Euclid. Though Euclid may have inspired the formal systems alluded to by Gowers, Euclid himself is not using a formal system, or at least he is not obviously using one, even though it may well be possible to retrofit the Elements with a formal system, as is done by Avigad, Dean, and Mumma []. Mathematics might be held up as an example of the peaceable resolu- tion of disputes. Gowers finds it to be unique in this way [, p. ]:

There is no mathematical equivalent of astronomers who still believe in the steady-state theory of the universe, or of biologists who hold, with great conviction, very different views about how much is explained by natural selection, or of philosophers who disagree fundamentally about the relationship between consciousness and the physical world, or of economists who follow opposing schools of thought such as monetarism and neo-Keynesianism. And yet, if all mathematical disputes are resoluble in principle, this is only because we accept the principle. Collingwood finds the same principle at work in history. “History,” he says [, p. ],   Philosophy of History

has this in common with every other science: that the historian is not allowed to claim any single piece of knowledge, except where he can justify his claim by exhibiting to himself in the first place, and secondly to anyone else who is both able and willing to follow his demonstration, the grounds upon which it is based. This is what was meant, above, by describing history as inferential. Mathematics is inferential in the same way. Now, can the grounds of an inference be accepted by one person, yet rejected by another? Practically speaking, it is less likely in mathematics than elsewhere. Nonetheless, it can happen anywhere, because one can always be faced with a skeptic, whom Collingwood goes on to distinguish from a proper critic:

a critic is a person able and willing to go over somebody else’s thoughts for himself to see if they have been well done; whereas a sceptic is a person who will not do this; and because you cannot make a man think, any more than you can make a horse drink, there is no way of proving to a sceptic that a certain piece of thinking is sound, and no reason for taking his denials to heart. It is only by his peers that any claimant to knowledge is judged.

Mathematical disputes are resoluble in principle. In practice, they may not be resoluble; but in this case, we may say of the parties to the dispute that one is a skeptic in Collingwood’s sense. This is possible in any dispute, be it in mathematics or history or anywhere else. According to Zeeman as quoted above, “The historian thinks extrinsi- cally in terms of the written evidence and adheres strictly to that data,” although this is admittedly a “traditional” view. It sounds like the ob- solescent view of history that distinguishes history from archeology and that is described by Collingwood as “scissors and paste”:

It is characteristic of scissors-and-paste history, from its least critical to its most critical form, that it has to do with ready-made statements, and that the historian’s problem about any one of these statements is whether he shall accept it or not: where accepting it means reasserting it as a part of his own historical knowledge. [, p. ] We can take an example from Euclid. Among the “definitions” at the head of Book vii of the Elements, there are the following two statements, . Evidence  numbered  and  in the Greek text established by Heiberg [], and hence so numbered in translations like Heath’s []: Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα. Μέρη δέ, ὅταν μὴ καταμετρῇ. A number is part of a number, the less5 of the greater, when it measures the greater. But parts, when it does not measure. In the body of Book vii, Proposition  has the following enunciation. ῞Απας ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη. Every number is of every number, the less of the greater, either a part or parts.

If the first two statements are really definitions, then the last statement follows immediately from them; but Euclid gives a nontrivial proof any- way. Suppose we think of geometry as a body of knowledge consisting of axioms and the theorems that they entail; and suppose we are, as it were, historians of this geometry. If we are scissors-and-paste historians, then we are going to reject at least one of the three ready-made statements of Euclid, because they cannot all be respectively definitions and theorem. In fact we are going to reject none one of the statements out of hand; nor shall we just accept them as definitions or theorem. We shall do something like what Collingwood goes on to describe:

Confronted with a ready-made statement about the subject he is study- ing, the scientific historian never asks himself: ‘Is this statement true or false?’, in other words ‘Shall I incorporate it in my history of that subject or not?’ The question he asks himself is: ‘What does this state- ment mean?’ And this is not equivalent to the question ‘What did the person who made it mean by it?’, although that is doubtless a question

Etymologically, “less” is a comparative form, although we seem not to have retained a positive form of its root, but we take “little” for the positive form []. The second S in “less” can be considered to stand for the R of the usual comparative suffix “-er.” The word “lesser” is thus a double comparative, as “greaterer” would be, if there were such a word.   Philosophy of History

that the historian must ask, and must be able to answer. It is equiv- alent, rather, to the question ‘What light is thrown on the subject in which I am interested by the fact that this person made this statement, meaning by it what he did mean?’ Our ultimate interest is in our own statements, as mathematicians and historians of mathematics. Collingwood describes these statements on his next page:

A statement to which an historian listens, or one which he reads, is to him a ready-made statement. But the statement that such a statement is being made is not a ready-made statement. If he says to himself ‘I am now reading or hearing a statement to such and such effect’, he is himself making a statement; but it is not a second-hand statement, it is autonomous. He makes it on his own authority. And it is this autonomous statement that is the scientific historian’s starting- point . . . If the scientific historian gets his conclusions not from the statement that he finds ready made, but from his own autonomous statement of the fact that such statements are made, he can get conclusions even when no statements are made to him. Thus the mathematician’s activity described by Zeeman, the rewriting of Euclid’s mathematics in one’s own notation, the better to under- stand it—this would seem to be an activity of the scientific historian, in Collingwood’s terms. “In scientific history” says Collingwood [, p. ],

anything is evidence which is used as evidence, and no one can know what is going to be useful as evidence until he has had occasion to use it.

Thus the mathematics that we work out can serve as evidence of what Euclid was doing. And yet it must be used with care, being based, as it is, on the presupposition of a kind of unity of mathematics—the presupposition about mathematical truths that, as Collingwood says, “if

Since we shall look briefly at Euclid’s use of the definite article in §. (p. ), let us note that the word “the” in “the better” is not the usual definite article descended from the Old English þe. It is rather a descendent of this pronoun’s instrumental case, spelled as þy and þon []. . Re-enactment  they are anywhere, they are everywhere, and if they are at any time they are at all times.” In the study of Euclid at least, it is desirable to question this presuppo- sition. In his article [] on the International Congresses of Mathematics, David Mumford’s theme is that the unity of mathematics (as reflected in the very existence of these Congresses) cannot be taken for granted, but must be worked for, and will not be achieved by forcing all of mathe- matics into the same mold. This is especially true when the mathematics under consideration is spread out over more than two thousand years.

. Re-enactment

In a quotation above from The Principles of History, Collingwood distin- guishes between what a statement means and what the person who made the statement means. We want to understand the mathematics thought about by Euclid, and we want to understand what Euclid thought about the mathematics; but these are different. Collingwood does not seem so concerned with the distinction in An Autobiography; but he does es- tablish there that all history is the history of thought. He distinguishes history from pseudo-history, the latter meaning

the narratives of geology, palaeontology, astronomy, and other natural sciences which in the late eighteenth and the nineteenth centuries had assumed a semblance at least of historicity . . . History and pseudo-history alike consisted of narratives: but in his- tory these were narratives of purposive activity, and the evidence for them consisted of relics they had left behind (books or potsherds, the principle was the same) which became evidence precisely to the extent to which the historian conceived them in terms of purpose, that is, understood what they were for. [, pp. –] Thus we shall ask of Euclid’s “definition” of part and parts of a number, and of his propositions about the same: what are they for?

I expressed this new conception of history in the phrase: ‘all history is the history of thought.’ You are thinking historically, I meant, when you say about anything, ‘I see what the person who made this (wrote   Philosophy of History

this, used this, designed this, &c.) was thinking.’ Until you can say that, you may be trying to think historically but you are not succeeding. And there is nothing else except thought that can be the object of historical knowledge. Political history is the history of political thought: not ‘political theory’, but the thought which occupies the mind of a man engaged in political work: the formation of a policy, the planning of means to execute it, the attempt to carry it into effect, the discovery that others are hostile to it, the devising of ways to overcome their hostility, and so forth. [, p. ]

As historians of mathematics, and in particular of Euclid’s mathematics, we study what Euclid was thinking while composing the Elements. We should acknowledge at some point that the author of the Elements, whom we call Euclid, was not necessarily one person. This should remind us that, if different parts of the Elements, or even of one of its thirteen books, do not seem to fit together, it may be because different hands put them together. Then again, we can decide the question of the value of the Elements for us, without knowing whether Euclid was one person or many. This is a point made by Collingwood in The Principles of Art of  [, pp. –]:

Individualism would have it that the work of a genuine artist is al- together ‘original’, that is to say, purely his own work and not in any way that of other artists . . . All artists have modelled their style upon that of others, used subjects that others have used, and treated them as others have treated them already . . . The individualistic theory of authorship would lead to the most ab- surd conclusions. If we regard the Iliad as a fine poem, the question whether it was written by one man or by many is automatically, for us, settled. If we regard Chartres cathedral as a work of art, we must contradict the architects who tell us that one spire was built in the twelfth century and the other in the sixteenth, and convince ourselves that it was all built at once.

If the Elements is a fine work of mathematics, its authorship is not nec- essarily singular; if not, not necessarily multiple. For Collingwood in An Autobiography, our study of Euclid is not only history, but typical or exemplary history. We want to understand the . Re-enactment  thoughts expressed in the Elements, and to do this, since they are math- ematical thoughts, we have to be mathematicians:

the historian must be able to think over again for himself the thought whose expression he is trying to interpret. If for any reason he is such a kind of man that he cannot do this, he had better leave that prob- lem alone. The important point here is that the historian of a certain thought must think for himself that very same thought, not another like it. If some one, hereinafter called the mathematician, has written that twice two is four, and if some one else, hereinafter called the historian, wants to know what he was thinking when he made those marks on paper, the historian will never be able to answer this question unless he is mathematician enough to think exactly what the mathematician thought, and expressed by writing that twice two are four. When he interprets the marks on paper, and says, ‘by these marks the mathe- matician meant that twice two are four’, he is thinking simultaneously: (a) that twice two are four, (b) that the mathematician thought this, too; and (c) that he expressed this thought by making these marks on paper. I will not offer to help a reader who replies, ‘ah, you are making it easy for yourself by taking an example where history really is the history of thought; you couldn’t explain the history of a battle or a political campaign in that way.’ I could, and so could you, Reader, if you tried. This gave me a second proposition: ‘historical knowledge is the re- enactment in the historian’s mind of the thought whose history he is studying.’ [, pp. –] Re-enactment is not discussed in The Principles of History, as we have the text; but re-enactment is discussed in some preliminary notes that Collingwood made for the book [, pp. –]:

all genuine historians interest themselves in the past just so far as they find in it what they, as practical men, regard as living issues. Not merely issues resembling these: but the same issues . . . And this must be so, if history is the re-enactment of the past in the present: for a past so re-enacted is not a past that has finished happening, it is happening over again. People have been quite right to say that the historian’s business is not to narrate the past in its entirety . . . but to narrate such of   Philosophy of History

the past as has historical importance . . . historical importance means importance for us. And to call a thing important for us means that we are interested in it, i.e. that it is a past which we desire to re-enact in our present. We are studying Euclid for the sake of doing mathematics today. Re- enacting Euclid means doing his mathematics. There are things called re-enactment that are not what Collingwood has in mind. It does not mean dressing in period costumes and aping archaic manners. Thus, the passage from An Autobiography mentioning political history continues: Consider how the historian describes a famous speech. He does not concern himself with any sensuous elements in it such as the pitch of the statesman’s voice, the hardness of the benches, the deafness of the old gentleman in the third row: he concentrates his attention on what the man was trying to say (the thought, that is, expressed in his words) and how his audience received it (the thoughts in their minds, and how these conditioned the impact upon them of the statesman’s thought). [, p. ] Treating the Elements historically does not mean just reading it, or re- producing its propositions in lectures delivered at a blackboard. It does involve understanding those propositions somehow. Historical study of Euclid does not even require learning Ancient Greek, though this does seem to be of value, and I shall usually quote Euclid in Greek as well as English. Euclid’s Greek is easy, being formulaic and having a small lexicon, many of whose words can be seen in the mathematical language of today. Proper re-enactment of the past is nonetheless a grand project. Many years before writing the books quoted so far, Collingwood engaged in another grand project, in Speculum Mentis of  [, p. ]: This book is the outcome of a long-growing conviction that the only philosophy that can be of real use to anybody . . . is a critical review of the chief forms of human experience . . . We find people practicing art, religion, science, and so forth, seldom quite happy in the life they have chosen, but generally anxious to persuade others to follow their example. Why are they doing it, and what do they get for their pains? This question seems, to me, crucial for the whole of modern life . . . . Re-enactment 

After using the bulk of the book to review the “forms of experience” called art, religion, science, history, and philosophy, Collingwood sums things up [, pp. –]:

We set out to construct a map of knowledge on which every legit- imate form of human experience should be laid down, its boundaries determined, and its relations with its neighbors set forth . . . Such a map of knowledge is impossible . . . Beginning, then, with our assumption of the separateness and au- tonomy of the various forms of experience, we have found that this separateness is an illusion . . . The various countries on our initial map, then, turn out to be vari- ously-distorted versions of one and the same country . . . What, then, is this one country? It is the world of historical fact, seen as the mind’s knowledge of itself.

I would read the emphasis on historical fact as follows. To know a the- orem, such as Fermat’s “Little” Theorem, means knowing that it is true, and this means knowing, as historical fact, that one has worked through a proof of the theorem and verified its correctness. One may have estab- lished a proof by induction, having noted that () 1p 1 (mod p), and () if ap a (mod p), then (a + 1)p ap + 1 a +1 (mod≡ p). One can then file≡ the proof away. Asserting the≡ theorem≡ as true does not require reopening the file; but it requires summoning up the historical fact of the proof’s having been placed in the file. One may find this “historical fact” inadequate; for, in the words of Paul Bahn quoted earlier (§., p. ), “nobody knows what happened in the past (even in the recent historical past).” If these words are to be taken seriously, then the memory of having proved a theorem does not suffice to provide knowledge that the theorem is true. Maybe we made a mistake when we proved the theorem yesterday. Maybe as students we accepted a teacher’s proof, but had not yet acquired sufficient mathematical skep- ticism to find the gaps in the proof. We should then try to ensure that the stating of a theorem includes within itself an actual proof. I think this is Mazur’s “self-proving theorem principle” [, p. ], namely:   Philosophy of History

if you can restate a theorem, without complicating it, so that its proof, or the essence of its proof, is already contained in the statement of the theorem, then you invariably have • a more comprehensible theorem, • a stronger theorem, and • a shorter and more comprehensible proof! For Mazur, the self-proving formulation of the Euclidean Algorithm is,

Suppose you are given a pair of numbers A and B with A greater than B. Any common divisor of A and B is a common divisor of B and A B; and conversely, any common divisor of B and A B is a − − common divisor of A and B.

This formulation gives us a key part of the theorem; but it omits that the process of continually replacing the greater of two numbers with the difference of the two numbers must eventually come to an end. That the natural numbers are well ordered must still be summoned up somehow. Collingwood seems to have anticipated our difficulty. As we were read- ing him in Speculum Mentis, he was speaking of a country, namely “the world of historical fact”:

Can we, then, sketch this country’s features in outline? We cannot. To explore that country is the endless task of the mind; and it only exists in its being explored. Of such a country there is no map, for it is itself its own map . . . There is and can be no map of knowledge, for a map means an abstract of the main features of a country, laid before the traveller in advance of his experience of the country itself. Now no one can describe life to a person who stands on the threshold of life. The maxims given by age to youth are valueless not because age means nothing by them but because what it means is just its own past life. To youth they are empty words. The life of the spirit cannot be described except by repeating it: an account of it would just be itself.

The bare statement of a theorem is thus a valueless maxim given by age to youth. This would appear to be so, even if “age” here is our younger self. As Wordsworth wrote in “My Heart Leaps Up” [, p. ], . Re-enactment 

The child is the father of the man; And I could wish my days to be Bound each to each by natural piety.

The child, or younger person, passes along a theorem to the man, or older person; but how can the older person properly receive this legacy? How can one day be remembered in the next, without being simply relived? To give a proper account of Euclid would seem to require us to repeat him; not to parrot him, but, as it were, to be him. The same goes for Collingwood too, by the way. If I parrot him here, in the sense of just quoting him, it is because I think his words need little translation, but usually make good-enough sense as they are. Further sense will come, if it comes at all, from applying them to one’s own experience, such as the experience of reading Euclid. What Serge Lang says of mathematics [, p. v] is true for any of Collingwood’s “forms of human experience”:

Unfortunately, a book must be projected in a totally ordered way on the page axis, but that’s not the way mathematics “is,” so readers have to make choices how to reset certain topics in parallel for themselves, rather than in succession.

Thoughts projected on the axis of time become totally ordered; but some- times a different ordering is needed for understanding them. My present anthology of quotations of Collingwood (and others) is intended to offer such an ordering. In any case, our main purpose is to understand Euclid; Collingwood is here to serve that purpose. On the other hand, if that purpose is served, this in turn will illuminate Collingwood. There is a certain pessimism or futility in Speculum Mentis: the old— Euclid—can give nothing to the young—us—, but we must go over ev- erything thoroughly for ourselves. And yet progress is possible:

In the toil of art, the agony of religion, and the relentless labour of science, actual truth is being won and the mind is coming to its own true stature. [, p. ]

Progress of a sort is possible:   Philosophy of History

This process of the creation and destruction of external worlds might appear, to superficial criticism, a mere futile weaving and unweaving of Penelope’s web, a declaration of the mind’s inability to produce solid assets, and thus the bankruptcy of philosophy. And this it would be if knowledge were the same thing as information, something stored in encyclopaedias and laid on like so much gas and water in schools and universities. But education does not mean stuffing a mind with information; it means helping a mind to create itself, to grow into an active and vigorous contributor to the life of the world. The information given in such a process is meant to be absorbed into the life of the mind itself, and a boy leaving school with a memory full of facts is thereby no more educated than one who leaves table with his hands full of food is thereby fed. At the completion of its education, if that event ever happened, a mind would step forth as naked as a new-born babe, knowing nothing, but having acquired the mastery over its own weaknesses, its own desires, its own ignorance, and able therefore to face any danger unarmed. [, p. ]

How does the mind “create itself,” with the help of education? Colling- wood continued to work on this question. The pessimism of Speculum Mentis did not lead to despair. Collingwood continued to think, as the mathematician continues to think after finding an error in a supposed proof. The answer given in An Autobiography is summarized in three “propo- sitions,” two of which we have seen: history is history of thought, and historical knowledge is re-enactment of thought. Then there is

my third proposition: ‘Historical knowledge is the re-enactment of a past thought incapsulated in a context of present thoughts which, by contradicting it, confine it to a plane different from theirs.’ How is one to know which of these planes is ‘real’ life, and which mere ‘history’? By watching the way in which historical problems arise. Every historical problem ultimately arises out of ‘real’ life. The scissors- and-paste men think differently: they think that first of all people get into the habit of reading books, and then the books put questions into their heads. But am not talking about scissors-and-paste history. In the kind of history that I am thinking of, the kind I have been practising all my life, historical problems arise out of practical problems. We study . Science 

history in order to see more clearly into the situation in which we are called upon to act. Hence the plane on which, ultimately, all problems arise is the plane of ‘real’ life: that to which they are referred for their solution is history. [, p. ]

To the question how we know which plane of life is real, and which history, I am not sure whether Collingwood’s answer is any clearer than saying, “We just do know.” Collingwood works through the example of Admiral Nelson, on the deck of the Victory, wondering whether to remove his decorations so as to become a less conspicuous target to snipers on enemy ships. To understand Nelson’s answer, as apparently Collingwood tried as a boy, we have to understand what Nelson thinks about the question; and yet we still know that the question does not actually arise in our own lives. Nonetheless, young Collingwood may have had a personal interest in the question, knowing that Nelson had a friend and colleague bearing the name of Collingwood, who took command at the Battle of Trafalgar after Nelson’s death []. We are interested in Euclid now, and difficulties in reading him arise precisely when the questions that underlie what he tells us are not our own questions. We must make them our own questions, and this will be re-enacting them. But the very difficulty of doing this will tell us that we are doing history. This is how I understand Collingwood’s “third proposition.”

. Science

Before continuing with Euclid, I want to note again that, for Colling- wood in The Principles of History, history is a science. This classifi- cation agrees with the general account of science given in An Essay on Metaphysics of :

The word ‘science’, in its historical sense, which is still its proper sense not in the English language alone but in the international language of European civilization, means a body of systematic or orderly thinking about a determinate subject-matter. This is the sense and the only sense in which I shall use it. [, p. ]   Philosophy of History

And yet earlier, in An Essay on Philosophical Method of , Colling- wood distinguished history from science:

Historical thought concerns itself with something individual, scientific thought with something universal; and in this respect philosophy is more like science than history, for it likewise is concerned with some- thing universal: truth as such, not this or that truth; art as such, not this or that work of art. In the same way exact science considers the circle as such, not this or that individual instance of it; and empirical science considers man as such, not, like history, this man as distinct from that. [, p. ]

In An Autobiography [, p. ], Collingwood described An Essay on Philosophical Method as

my best book in matter; in style, I may call it my only book, for it is the only one I ever had the time to finish as well as I knew how, instead of leaving it in a more or less rough state.

Did Collingwood nonetheless change his mind about science later? The passage from An Essay on Metaphysics continues:

There is also a slang sense of the word [‘science’], unobjectionable (like all slang) on its lawful occasions, parallel to the slang use of the word ‘hall’ for a music-hall or the word ‘drink’ for an alcoholic drink, in which it stands for natural science.

In the earlier Essay on Philosophical Method, was Collingwood using the word “science” in its slang sense? Does the term “natural science” in the later Essay encompass both empirical and exact science? These questions are not of great importance for us, but provide an opportunity to note the grand theme of the earlier Essay, which is the “overlap of classes”:

The specific classes of a philosophical genus do not exclude one another, they overlap one another. This overlap is not exceptional, it is normal; and it is not negligible in extent, it may reach formidable dimensions.

Thus in The Principles of Art of , Collingwood will be at pains to distinguish art from craft, although, for example, . Science 

The distinction between planning and executing certainly exists in some works of art, namely those which are also works of craft or artifacts; for there is, of course, an overlap between these two things, as may be seen by the example of a building or a jar, which is made to order for the satisfaction of a specific demand, to serve a useful purpose, but may none the less be a work of art. But suppose a poet were making up verses as he walked . . . [, p. ] The poem may be unplanned, and so not be craft, but nonetheless be a work of art; yet other things are both craft and art. The overlap of classes has a practical result [, p. ]. First:

On a matter of empirical fact it is possible, when asked for example ‘where did I leave my purse?’ to answer ‘not in the taxi, I am sure’, without having the least idea where the purse was actually left... Likewise, when asked, “Is this proof correct?” we may answer “No,” with- out having the least idea of a correct proof, if indeed there is a correct proof. Nonetheless,

In philosophy this is not so. The normal and natural way of replying to a philosophical statement from which we dissent is by saying, not simply ‘this view seems to me wrong’, but ‘the truth, I would suggest, is something more like this’, and then we should attempt to state a view of our own . . . This is not a mere opinion. It is a corollary of the Socratic principle (itself a necessary consequence of the principle of overlapping classes) that there is in philosophy no such thing as a transition from sheer ignorance to sheer knowledge, but only a progress in which we come to know better what in some sense we know already. Euclid’s Elements is not philosophy; and yet perhaps the best reason for reading it is philosophical: to deepen our understanding of mathematics as we already know it. In this case, if we detect what we think are errors in Euclid, we ought to be prepared to correct them, and correct them in a way that Euclid himself would agree with. Mathematics is not philosophy; but there is an overlap. After writ- ing An Essay on Philosophical Method, with its doctrine of the overlap of classes, Collingwood applied the method to nature; the resulting lec- tures were posthumously published in  as The Idea of Nature. What   Philosophy of History

Collingwood says at the beginning about natural science applies as well to mathematics:

The detailed study of natural fact is commonly called natural science, or for short simply science; the reflection on principles, whether those of natural science or of any other department of thought or action, is commonly called philosophy. Talking in these terms, and restricting philosophy for the moment to reflection on the principles of natural science, what I have just said may be put by saying that natural sci- ence must come first in order that philosophy may have something to reflect on; but that the two things are so closely related that natural science cannot go on for long without philosophy beginning; and that philosophy reacts on the science out of which it has grown by giving it in future a new firmness and consistency arising out of the scientist’s new consciousness of the principles on which he has been working. For this reason it cannot be well that natural science should be as- signed exclusively to one class of persons called scientists and philoso- phy to another class called philosophers. [, p. ]

Collingwood saw the need for a bridge between science and philosophy, and thought himself not alone in this:

In the nineteenth century a fashion grew up of separating natural scien- tists and philosophers into two professional bodies, each knowing little about the other’s work and having little sympathy with it. It is a bad fashion that has done harm to both sides, and on both sides there is an earnest desire to see the last of it and to bridge the gulf of misunder- standing it has created. The bridge must be begun from both ends; and I, as a member of the philosophical profession, can best begin at my end by philosophizing about what experience I have of natural science. Not being a professional scientist, I know that I am likely to make a fool of myself; but the work of bridge-building must go on. [, p. ]

Why build bridges? Collingwood had seen the great destructive folly of the First World War, albeit from a desk in London at the Admiralty Intelligence Division [, p. ].

The War was an unprecedented triumph for natural science. Bacon had promised that knowledge would be power, and power it was: power . Science 

to destroy the bodies and souls of men more rapidly than had ever been done by human agency before. This triumph paved the way to other triumphs: improvements in transport, in sanitation, in surgery, medicine, and psychiatry, in commerce and industry, and, above all, in preparations for the next war. But in one way the War was an unprecedented disgrace to the human intellect . . . nobody has ever supposed that any except at most the tiniest fraction of the combatants wanted it . . . I seemed to see the reign of natural science, within no very long time, converting Europe into a wilderness of Yahoos. [, pp. –]

Collingwood saw salvation in history, pursued scientifically in the sense described earlier, and not by scissors and paste. It was suggested earlier that a kind of salvation from conflict lay in mathematics, where all disputes could be resolved amicably. And yet, strictly speaking, what can be so resolved are disputes about mathematics that can be expressed in formal systems. Mathematics, and number theory in particular, can no longer be distinguished from other sciences as Hardy distinguished it in  by saying,

science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. [, §, p. ]

The Elements is an old book, and certain old books have been, and continue to be, the nominal causes of bloody disputes. Euclid is different. There have been no wars in his name. There may still be academic disputes about him. If history has the potential that Collingwood saw in it, Euclid may be as good a place as any for a mathematician, at least, to try out the possibilities.  Euclid in History

. Dedekind and Hilbert

Today we may be better able to learn Euclid’s mathematics, precisely because the Elements is not commonly used as a textbook. We can approach Euclid more easily now, without assuming that he is doing just what we are doing when we do mathematics. In the preface of his  translation of the Elements [, v. i, p. vii], Thomas Heath said,

no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers.

I do not know why engineers and schoolchildren, boys and girls, do not also deserve to know the real Euclid; but in any case, I suppose mathe- maticians of Heath’s day did know Euclid. Whether they knew him more directly than through the “revised or rewritten versions” that Heath refers to, I do not know. However, a few years before Heath, David Hilbert in- troduced his Foundations of Geometry [, p. ] by saying,

Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry. The choice of the axioms and the investigation of their relations to one another is a problem which, since the time of Euclid, has been discussed in numerous excel- lent memoirs to be found in the mathematical literature. This problem is tantamount to the logical analysis of our intuition of space.

Presumably Hilbert had actually read Euclid, at least in some form. Thus he could disagree explicitly with Euclid on whether the congruence of all

 . Dedekind and Hilbert  right angles is an axiom or a theorem [, p. ]. On the other hand, this disagreement may represent a somewhat careless reading of Euclid; I shall discuss this point in §. (p. ). A few years before Hilbert, Richard Dedekind [, pp. –] com- plained about critics who thought that his theory of irrational numbers could be found already in the Traité d’Arithmétique of Joseph Bertrand. According to Dedekind, the theory in Bertrand’s work had already been present in Euclid’s work; but his own theory was different. Dedekind traced his own definition of irrational numbers to the idea that an irrational number is defined by the specification of all rational num- bers that are less and all those that are greater than the number to be defined . . . That an irrational number is to be considered as fully de- fined by the specification just described, this conviction certainly long before the time of Bertrand was the common property of all mathe- maticians who concerned themselves with the irrational . . . [I]f, as Bertrand does exclusively in his book (the eighth edition, of the year , lies before me,) one regards the irrational number as the ratio of two measurable quantities, then is this manner of determining it al- ready set forth in the clearest possible way in the celebrated definition which Euclid gives of the equality of two ratios (Elements, V., ). In the  edition of his Traité [] (the one that I was able to find), Bertrand indeed defines irrational square roots and other irrational num- bers as measures of quantities with respect to a predetermined unit. Thus he writes: . Lorsqu’un nombre N, n’est le carré d’aucun nombre entier ou fractionnaire, la définition de sa racine carrée exige quelques développe- ments. On dit qu’un nombre est plus grand ou plus petit que √N suivant que son carré est plus grand ou plus petit que N; d’après cela, pour définir les grandeurs dont √N est la mesure, supposons, par exemple, qu’après avoir adopté une certaine unité de longueur on regarde tous les nombres comme exprimant des longueurs portées sur une mème ligne droite à partir d’une origine donnée. Une portion de cette ligne recevra les extrémités des longueurs dont la mesure est moindre que √N, et une autre portion celles des lignes dont la mesure est plus grande que √N; entre ces deux régions, il ne pourra évidemment exister aucun   Euclid in History

intervalle, mais, seulement, un point de démarcation. La distance à laquelle se trouve ce point, est, par définition, mesurée par √N. That is, if N is not a square, we define √N by saying first x > √N if x2 > N, and x < √N if x2 < N, where implicitly the x are rational. If such x are taken to measure lengths along a straight line from a given origin, a unit length having been chosen, then the extremities of the lengths measuring less than √N will be separated from the extremities of the lengths measuring more than √N by a single point, whose distance from the origin is √N. Bertrand’s account has some obscurities, such as how a length can be distinguished from the bounded straight line of which it is a length. Dedekind seems right in saying that Euclid had “already set forth in the clearest possible way” the approach that Bertrand would take. Thus, suppose A and B are magnitudes that have a ratio to one another in the sense of Definition v. of the Elements. This means some multiple of either magnitude exceeds the other, or, in our terms, A and B are positive elements of an archimedean ordered group. We derive from these magnitudes the set m/n: mB < nA consisting of positive rational numbers; here m and n range{ over the set }N of positive integers. The set m/n: mB < nA determines the ratio of A to B, in the sense that, if {C and D are also} magnitudes having a ratio to one another, then they have the same ratio that A and B have, provided

m/n: mB < nA = m/n: mD < nC . { } { } Such is Euclid’s Definition v., in modern form. I suggested in §. (p. ) that fractions were problematic; but in the present context, we can just replace m/n with the ordered pair (m, n). Dedekind’s insight was that a set of the form m/n: mB < nA had properties that could be specified without reference{ to magnitudes like} A and B; and then the sets of rational numbers with those properties could

In the paper [] discussed earlier, Zeeman shows how to avoid using the full group structure in order to define the ratio of A to B. One may not be able to find the difference of A and B; but if A

then is the space made up of the points M, as is easy to see, every- where discontinuous; but in spite of this discontinuity, and despite the existence of gaps in this space, all constructions that occur in Euclid’s Elements, can, so far as I can see, be just as accurately effected as in perfectly continuous space; the discontinuity of this space would not be noticed in Euclid’s science, would not be felt at all. If any one should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides ra- tional quantitative relations, also irrational, and besides algebraic, also transcendental quantitative relations are conceivable.

Mathematicians had drifted away from the rigor of Euclid without re- alizing it. They worked with kinds of numbers never contemplated by Euclid, while still founding their ideas of such numbers in geometric in- tuition. For Dedekind, this would not do. There is more to the story; but the point for now is that when there has been a continuous tradition of building up mathematics from Euclid, the tradition may lose sight of what Euclid actually did. Now that we no longer have this tradition, we

My article [] has some developments from Descartes’s idea.   Euclid in History may be better able to do what Dedekind could do, and understand what Euclid really did do.

. From a course of Euclid

Some students do still learn mathematics from Euclid. My own math- ematics department in Turkey now has a first-semester undergraduate course based on the first book of the Elements. Students go to the board and demonstrate propositions, more or less on the pattern of my own alma mater, St. John’s College in the United States. However, they use a Turkish translation of Euclid prepared in collaboration with my col- league Özer Öztürk []. The translation is from the original Greek, as established by Heiberg []. One could just translate Heath’s English []; but there are ways that Heath is inaccurate. In translating the first proposition of the first book of the Elements, Heath begins, On a given finite straight line to constuct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construct an equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described. But what Euclid says (in Heiberg’s transcription and in my literal trans- lation) is, ᾿Επὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συς- τήσασθαι. ῎Εστω ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ. ∆εῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι. Κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓ∆. // [1] On the given finite straight [line] to construct an equilateral triangle. [2] Let the given finite straight [line] be ΑΒ. [3] Thus it is required on the straight [line] ΑΒ to construct an equilateral triangle. [4] With center Α and distance ΑΒ let a circle have been drawn, ΒΓ∆. The four sentences here are, respectively, the four parts of a proposition that Proclus [, p. ] calls () enunciation, () exposition (or “setting out”), () specification, and () construction. (The fourth sentence is only part of the construction.) The remaining two parts of a proposition . From a course of Euclid 

Γ

∆ Α Β Ε

Figure .. Proposition i. are the () demonstration and () conclusion. In Proposition i., the construction continues with the drawing of a second circle as in Figure ., and with the connection of one of the points of intersection of the two circles with the endpoints of the original finite straight line. Then follows the (easy) demonstration that the resulting triangle is indeed equilateral, and the conclusion that what was to be done has been done. The literal translation of the Greek above shows five differences from Heath: . In the enunciation, Euclid refers not to a straight line, but the straight line. The definite article “the” here can be understood as be- ing generic, as in Wordsworth’s verse, “The child is the father of the man.” I discuss this matter in more detail elsewhere []. The straight line of Euclid’s enunciation can also be understood as the straight line in Euclid’s diagram, which exists before we start to read the proposition. . Today if we say that we are going to construct an equilateral triangle on a straight line, we mean an arbitrary straight line; but then we proceed to draw our own straight line, assigning to its endpoints, as Heath does, the letters A and B. This is not what Euclid does, as Reviel Netz explains in The Shaping of Deduction in Greek Mathematics [, pp. –]: Nowhere in Greek mathematics do we find a moment of specification per se, a moment whose purpose is to make sure that the attribution of letters in the text is fixed. Such moments are very common in mod- ern mathematics, at least since Descartes. But specifications in Greek   Euclid in History

mathematics are done, literally, ambulando. The essence of the ‘im- perative’ element in Greek mathematics—‘let a line be drawn . . . ’, etc.—is to do some job upon the geometric space, to get things moving there . . .  What we see, in short, is that while the text is being worked through, the diagram is assumed to exist. The text takes the diagram for granted.

Thus, at the start of Euclid’s proposition, we have already been given a straight line, labelled as ΑΒ. The exposition or “setting out” of the proposition tells us to understand the straight line of the enunciation as the straight line ΑΒ. We are not told to let ΑΒ be some straight line that we proceed to create for ourselves. . Heath tells us to let a circle be drawn. Using chalk or a pen, we can then draw a circle. But apparently Euclid had nothing like our blackboards or whiteboards, with which a diagram could be constructed during a lecture. Again, his diagrams would already have been drawn, perhaps on a wax tablet. Referring to the last passage quoted, Netz says,

This, in fact, is the simple explanation for the use of perfect imperatives in the references to the setting out—‘let the point A have been taken’. It reflects nothing more than the fact that, by the time one comes to discuss the diagram, it has already been drawn.

Though seen in the command “Have done with it,” the English imperative in the perfect aspect is awkward. English does not even have a third- person imperative in any aspect, except in some formulas like “God bless you”; otherwise we achieve the same effect periphrastically, by means of the second-person imperative “let,” as in “Let it be done.” Turkish, like Greek, does have third-person imperatives, which are commonly used:

The first ellipsis here is Netz’s; the second, mine. “Periphrasis & civilisation are by many held to be inseparable; these good people feel that there is an almost indecent nakedness, a reversion to barbarism, in saying No news is good news instead of The absence of intelligence is an indication of satisfactory developments. Nevertheless, The year’s penultimate month is not in truth a good way of saying November.” H. W. Fowler, A Dictionary of Modern English Usage [, Periphrasis, p. ]. . From a course of Euclid 

let somebody draw γραφέτω çizsin let it be drawn γραφέσθω çizilsin let it have been drawn γέγραφθω çizilmiş olsun This similarity between Turkish and Greek was a particular reason to translate directly from Euclid’s Greek into Turkish. . For Euclid, a line is what we call a curve; our lines are Euclid’s straight lines. Heath is faithful to Euclid by writing “straight line” when this is what is meant. However, Euclid usually abbreviates “straight line” (εὐθεῖα γραμμή) to “straight” (εὐθεῖα). Heath cannot do this if, as he does, he wants to maintain good English style. One can do it in Turkish though: here a line in Euclid’s sense is çizgi, while a straight line is doğru çizgi or just doğru. . The foregoing differences between Heath and Euclid are perhaps of little mathematical importance. A fifth difference brings out a remarkable feature of Euclid. Heath italicizes the enunciations of Euclid’s proposi- tions; but Euclid had no such means of emphasizing text. He did not even have the medieval distinction between minuscule and capital letters. Like Heath, Heiberg emphasizes Euclid’s enunciations: not however by a change of font, but by spacing out the letters like this. I assume Euclid did not consider doing such a thing, since he would not even have separated words with spaces, butwrotecontinuouslylikethis. I suppose there still would have been the possibility of underlining for emphasis. Any chance In The Mathematics of Plato’s Academy [, §.], David Fowler dis- that underlines cusses what we know about ancient manuscript style. He also [, §.] were used? looks at what he calls the protasis-style of Euclid. Πρότασις is the term of Proclus that we translate as “enunciation.” Before Euclid, we have no evidence of any mathematics written in the Euclidean style, with an enunciation followed by justification. Aristotle’s Prior Analytics might be an exception, except that it is not really mathematics, but logic. To-

At the beginning of the Prior Analytics, Aristotle defines πρότασις as “an affirmative or negative statement of something about some subject” [, a, p. ]. We then translate the word as premiss or just proposition. Πρότασις is used in Greek today for an entire proposition of Euclid [], but I am not aware of any ancient basis for this usage.   Euclid in History day the protasis-style is ubiquitous in mathematics; and yet we signal our protases with rubrics like “Theorem N,” and our justifications with the word “Proof” (and a box at the end). Strictly speaking, this is not Euclid’s style. It is often not students’ style either. In performing the task of demon- strating something, students will write down various statements, without being clear about the logical relations between them. Even professional mathematicians will do this at the board, expecting the attentive listener to know what is meant (and expecting all listeners to be attentive). At least Euclid establishes a set pattern for his propositions. Read a bit, and you see that the Elements is a sequence of assertions, each followed by a justification, the justification itself following (more or less) the outline given by Proclus. Actually, some of Euclid’s enunciations are not assertions, but tasks, as in the very Proposition i. that we have been looking at. The distinction between an assertion and a task can be indicated by the labels theorem (θεόρημα) and problem (πρόβλημα). However, according to Pappus, “among the ancients some described them all as problems, some as theorems” [, p. ]. Euclid himself did not use such labels at all. Heath helps the reader typographically, by italicizing enunciations, by breaking the text into short paragraphs, and by centering some phrases. This typography may be misleading, if it causes us to think of Euclid’s propositions just as if they were modern theorems with proofs. In some cases at least, Euclid’s propositions are not like that. An example men- tioned above in §. (p. ), and to be further considered in §§. & . (pp.  & ) is Proposition vii..

. Equality

Another of Euclid’s propositions that is not like a modern theorem is Proposition i.. This is where Euclid establishes the principle of triangle congruence that we call “Side Angle Side.” For Hilbert, this principle is an . Equality 

Α ∆

Β Γ ΕΖ

Figure .. Proposition i. axiom, the one that he numbers IV, . Euclid proves the principle, but without using any postulate or any previous proposition. Can Euclid’s proof then be a “real” proof? Given triangles ΑΒΓ and ∆ΕΖ as in Figure ., let us suppose ΑΒ = ∆Ε and ΑΓ = ∆Ζ and ∠ ΒΑΓ = ∠ Ε∆Ζ. If we apply triangle to triangle so that Α falls on ∆, and ΑΒ falls along ∆Ε, then Β will fall on Ε, and ΑΓ will fall along ∆Ζ, so that Γ will fall on Ζ; and then ΒΓ will coincide with ΕΖ. So argues Euclid. Do we accept it? Students are sometimes disturbed by an expression like ΑΒ = ∆Ε; they want to make it ΑΒ = ∆Ε . Apparently they think that the sign of equality in fact denotes| | identity.| | Obviously the straight lines ΑΒ and ∆Ε are not identical; but they may have identical lengths, which can then be denoted indifferently by ΑΒ or ∆Ε . | | | | Euclid does not use a symbol for equality; he just says ΑΒ is equal (ἴσος) to ∆Ε. Along with definitions and postulates, the preamble of the Elements contains so-called Common Notions; and of the five of these that Heiberg and Heath accept as genuine, four establish what equality

More precisely, Hilbert’s axiom is that if two sides and the included angle of a triangle are respectively congruent to two sides and the included angle of another triangle, then the remaining angles are respectively congruent to the remaining angles. That the remaining side is congruent to the remaining side is his Theorem . More precisely ἴση, in agreement with the feminine gender of γραμμή; but the lemma or dictionary-form of the word is ἴσος.   Euclid in History

Figure .. Proposition i.

Figure .. Proposition i.

means. According to the assigned numbering, these Common Notions are that () things congruent to one another are equal to one another, and () things equal to the same thing are equal to one another; but moreover, if equals be () added to or () subtracted from equals, the results are equal. In Common Notion , I use the word congruent for Euclid’s participle ἐφαρμόζων, to emphasize that the notion of congru- ence found in Hilbert originates in Euclid’s notion of equality; but Heath says “things which coincide with” instead of things congruent to. For Euclid, figures are equal if congruent parts can be added or subtracted Motivic so as to obtain congruent figures. Thus parallelograms on equal bases integration? and in the same parallels can be shown to be equal to one another by cutting out and rearranging the parts shown in Figure .. The equality of these parallelograms is Proposition i.. Actually Euclid derives this equality from Proposition i., where the bases of the parallelograms are not equal, but the same (ὁ αὐτός), as in Figure .. Here the proof is by cutting and pasting; then i. is proved by means of a third parallelo- gram, which shares a base with either of the first two parallelograms, as in Figure .. In the Declaration of Independence of the United States of America, . Equality 

Figure .. Proposition i. as Euclid does it when Thomas Jefferson wrote the self-evident truth, “that all men are created equal” [, p. ], he did not mean that all persons were the same person. However, in mathematics today, we confuse equality and identity. Thus when I wrote the equation nx = my earlier, I meant that nx and my were to be considered as the same element of the set N, although I would read the sign = as “equals.” In the article called “When is one thing equal to some other thing?” [], the notion of equality that Barry Mazur contemplates is not distinct from the notion of sameness. Thus the article begins:

One can’t do mathematics for ten minutes without grappling, in some way or other, with the slippery notion of equality. Slippery, because the way in which objects are presented to us hardly ever, perhaps never, immediately tells us—without further commentary—when two of them are to be considered equal. We even see this, for example, if we try to define real numbers as decimals, and then have to mention aliases like 20 = 19.999 ... , a fact not unknown to the merchants who price their items $19.99. The heart and soul of much mathematics consists of the fact that the “same” object can be presented to us in different ways. Even if we are faced with the simple-seeming task of “giving” a large number, there is no way of doing this without also, at the same time, giving a heft amount of extra structure that comes as a result of the way we pin down—or the way we present—our large number. If we write our number as 1729 we are, sotto voce, offering a preferred way of “computing it” (add one thousand to seven hundreds to two tens to nine). If we present it as 1+123 we are recommending another mode of computation, and if we pin it down—as [Ramanujan] did—as the first number expressible as a sum of two cubes in two different ways, we are   Euclid in History

Α

Β Γ Ζ Η ∆ Ε

Figure .. Proposition i.

being less specific about how to compute our number, but we have underscored a characterizing property of it within a subtle diophantine arena.

When we are presented with the angles ΒΑΓ and Ε∆Ζ in the triangles in Figure ., there is nothing about this presentation itself that tells us that the two angles are equal, just as there is nothing about the two expressions 20 and 19.999 ... that tells us they stand for equal numbers. However, even when the expressions are understood, the equal angles ΒΑΓ and Ε∆Ζ are not interchangeable in the way that 20 and 19.999 ... are. We say that the numbers 20 and 19.999 ... are the same number; but ΒΑΓ and Ε∆Ζ are different angles of different triangles. In fact they could be the same angle, as they are for example in Propo- sition i., whose diagram is in Figure .. Here the sides ΑΒ and ΑΓ of the triangle ΑΒΓ are given as being equal to one another. By construction, ΑΖ = ΑΗ. Since also the angle ΖΑΗ is common to the triangles ΑΖΓ and ΑΗΒ, these two triangles are congruent to one another, by Proposition i.—but strictly speaking, this conclusion requires us to recognize that the common angle of the two triangles is equal to itself. We must also recognize that this angle can be expressed indifferently as ΖΑΓ or ΗΑΒ. Euclid does not say this explicitly. Mazur’s concern is more with the question of whether your triangles are the same as my triangles, or your numbers are the same as my numbers. . Equality 

He says [, p. ], “Equivalence (of structure) in the above ‘compromise’ is the primary issue, rather than equality of mathematical objects”— where again I think “equality” can be read as sameness. The “compromise” is between the treatment of the number 5 as a particular standard five- element set, and its treatment as the class of all five-element sets. The compromise is to let you use your five-element set, and let me use mine, as long as what each of us does with it can be “translated” into what the other does with it. Mazur elaborates on how the language of category theory lets us talk about these things. However, I think there is no question that, in any instance of actually doing mathematics, there is only one number 5. There may be different five-element sets, but there is only one five, be it a particular five-element set reserved as a standard, or be it the unique class consisting precisely of all five-element sets. For Euclid, the matter is less clear. But before looking at his treatment of numbers in particular, I want to answer the question raised at the beginning of this section. Euclid’s proof of Proposition i. is a real proof, because it is based on several applications of the principle that equal straight lines can be made to coincide, and likewise equal angles. This principle is not explicitly stated; but should it have been? What else can equality of straight lines or angles mean? The status of Proposition i., “Side Side Side,” is not so clear. In Figure ., if we now let ΑΒ = ΑΓ and ΑΓ = ∆Ζ and ΒΓ = ΕΖ, then we can apply the base ΒΓ to the base ΕΖ, since they are equal. Euclid argues that Α must fall on ∆, because otherwise it falls on a point Ζ as in Figure ., and then a contradiction to Proposition i. arises. If Euclid allows such an argument, perhaps he will allow Hilbert’s proof that all right angles are equal to one another. Still, perpendiculars can be erected and dropped (as in Propositions i. and ) without recourse to Proposition i., and then this proposition can be proved by means of perpendiculars, Proposition i., and the equality of all right angles. There is no reason not to think Euclid was aware of this, but preferred to leave such a proof as an exercise for the reader. Euclid does not say he is doing this; but then there is a lot that he does not spell out, as we shall discuss in §. (p. ).   Euclid in History

∆ Η

ΕΖ

Figure .. Proposition i.

. Right angles

We do not know what the preamble of the Elements consisted of when this collection of thirteen books was first compiled. In The Forgotten Revolution [, pp. –], Lucio Russo argues that Euclid’s obscure definition of straight line is only a later addition to the Elements: in origin it is a truncated sentence from a student’s crib-sheet. The definition in the Elements is Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ᾿ ἑαυτῆς σημείοις κεῖται, which is practically the same as the first part of Εὐθεῖα μὲν οὖν γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐπ´ αὐτῆς σημείοις κεῖται ὀρθὴ οὖσα καὶ οἷον ἐπ´ ἄκρον τεταμένη ἐπὶ τὰ πέρατα. Russo noticed this among the Definitions of Terms in Geometry at- tributed to Hero [, p. , ll. –]; it means something like A straight line is one that equally with respect to all points on itself lies right and maximally taught between its extremities. The italicized part is what is missing from the Elements, though it is where the meaning of the definition lies; the reference to “all points on

Russo’s translation, as rendered in English by Silvio Levy, is “A straight (εὐθεῖα) line is a line that equally with respect to all points on itself lies straight (ὀρθή) and maximally taught between its extremities.” . Right angles  itself” allows for the case of an unbounded straight line, where any two points must serve as extremities for the purpose of the definition. Here I just want to assert the plausibility of Russo’s argument. There is reasonable doubt about the authenticity of the definition of straight line in the Elements. Some understanding of straightness is needed for the conclusion in Proposition i. that, when points Β and Γ are made to coincide with points Ε and Ζ, then straight line ΒΓ will coincide with straight line ΕΖ. Whether Euclid meant this understanding to be part of Postulate , or part of the definition of straight line, or even just part of Proposition i., I do not know. The definition in the Elements of a right angle (ὀρθὴ γωνία) is given as,

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

This is Heath’s translation, with his italics. As the italics suggest, surely it is right angles that are being characterized, and not equality of angles. Equality of angles is implicitly understood; and indeed, what else can it mean but that one angle can be picked up and placed on another? Postulate  is that all right angles are equal to one another. I take this to mean that we have a toolkit containing a carpenter’s square. The square is not for drawing right angles: this is achieved by Propositions i. and . But the carpenter’s square reminds us that indeed all right angles are equal to one another, because they can be made to coincide with this one standard angle. (I take an actual carpenter’s square to my own Euclid classes, to serve as such a reminder.) Euclid’s first three postulates are that we can do things that can be done with a straightedge and compass, or just with a string or cord. We can () connect two points with a straight line, () extend a given straight line, and () draw a circle with given center, passing through another given point. Finally, Postulate  tells us something else that we can do: we can find a point of intersection of two straight lines, if we extend them far enough, provided that a line falling across them makes   Euclid in History

Α Ε Κ

∆ Β Γ Θ ΖΗ

Figure .. Postulate  the interior angles on the same side less than two right angles. Here the angles are together less than two right angles, though one of them might be greater than a right angle. Thus we are going to need a way to bring two angles together, for comparison with two right angles. I noted in §. (p. ) that Hilbert thought Euclid’s Postulate  was actually a theorem. In the style of Euclid, Hilbert’s proof would seem to be as follows. Suppose, as in Figure ., straight line ΑΒ is perpendicular to Γ∆, and ΕΖ is perpendicular to ΗΘ. If right angles ΑΒΓ and ΕΖΗ are not equal to another, then one is greater. Suppose the latter is greater. Then ΑΒΓ will fall inside it as ΚΖΗ. The supplement ΚΖΘ of ΚΖΗ must be equal to the supplement ΑΒ∆ of ΑΒΓ, though this needs further discussion: it is Hilbert’s Theorem . Absurdity results, since in short

∠ ΑΒΓ = ∠ ΚΖΗ < ∠ ΕΖΗ = ∠ ΕΖΘ < ∠ ΚΖΘ = ∠ ΑΒ∆ = ∠ ΑΒΓ.

How did we obtain angle ΚΖΗ? Hilbert can use his axiom IV, , which he summarizes as being

that every angle in a given plane can be laid off upon a given side of a given half-ray in one and only one way. For Euclid, this is a theorem, namely Proposition i.. Hilbert’s toolkit contains a protractor, or else triangles with angles of all posssible sizes; and these can be used to draw with. But again, the only triangles in Euclid’s toolkit are right triangles, and they cannot be used to draw with (and their acute angles cannot be used in any way). In Euclid’s Proposi- tion i., we are able to “lay off” one given angle on another, because this possibility is implicit in the assumption that the two angles are equal to . Right angles  one another. In Figure ., there is no assumed equality of anything on the left with anything on the right; so for Euclid, there can be no “laying off.” If he had wished to, Hilbert could have assumed ΑΒ = ΕΖ, and likewise for the other corresponding straight lines, because of his axiom IV, , summarized as being

that every segment can be laid off upon a given side of a given point of a given straight line in one and only one way.

This is Euclid’s Proposition i.. There must have been a great change in thinking about mathematics, if Hilbert was trying to build up geometry on a minimal foundation, and yet could treat as axiomatic what for Euclid needed proof. Hilbert seems not to have been troubled by this change. Perhaps this is because he was reading Euclid according to a tradition whose changes to the meaning of Euclid had gone unnoticed.  Euclid’s Foundations of Arithmetic

. Unity

Not only is the definition of straight line plausibly held to be a late addition to the Elements, but Russo says the same of the definition of unity, at the head of Book vii. His reason is that, six centuries after Euclid, Iamblichus described the same definition as being due to “more recent authorities.” Heath discusses the same passage of Iamblichus, both in his edition of the Elements [, v. ii, p. ] and in his History of Greek Mathematics [, p. ]. The discussion is more brief in the latter, where the very part of the passage that is relevant for us now is not considered. In his notes on the Elements, Heath gives no explicit indication that the authenticity of the definition of unity is called into question by the words of Iamblichus. The definition in Euclid is, Μονάς ἐστιν, καθ᾿ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται. Unity is that according to which each entity is said to be one thing. The translation is mine; Heath’s is “A unit is that by virtue of which each of the things that exist is called one.” I propose to try out unity instead of Heath’s “unit,” because the latter is a made-up word, albeit made up precisely to translate Euclid’s μονάς. In his preface to Billingsley’s English translation of the Elements, published in , John Dee wrote,

Number, we define, to be, a certayne Mathematicall Summe, of Vnits. And, an Vnit, is that thing Mathematicall, Indiuisible, by participation of some likenes of whose property, any thing, which is in deede, or is counted One, may resonably be called One.

 . Unity 

As Billingsley’s was the first English translation of the Elements [, v. , p. ], so is the passage from Dee the first of the illustrative quotations in the Unit article of the Oxford English Dictionary []. In the etymology section of that article, Dee’s marginal note—apparently on the passage above—is quoted: Note the worde, Vnit, to expresse the Greke Monas, and not Vnitie: as we haue all, commonly, till now, vsed. However, at Unity, the OED gives Billingsley’s translation of Euclid’s definition: Vnitie is that, whereby euery thing that is, is sayd to be on. Evidently Billingsley did not perceive a need for Dee’s new word “unit.” Billingsley and Dee could have used “monad” (from the stem μονάδ- of μονάς): this is now an English word, though its earliest quotation in the OED is from . The French monade dates to  [, p. ]. Mean- while, French is content to use the old unité where English puts the new- fangled “unit”: in the quotation from Bertrand’s Traité d’Arithmétique in §. (p. ), “unité de longueur” serves where we say “unit of length.” By the way, it is not clear whether Euclid’s μονάς and ἕν (“one”) are etymologically related. The Greek cardinal numeral “one” is declined as an adjective and, as such, has three distinct genders: the masculine, fem- inine, and neuter are respectively εἷς, μία, ἕν. Despite appearances, these three forms are related to one another: each comes from σεμ, according to Smyth’s old Greek Grammar [, ¶] and the more recent Chantraine [, II, , εἵς]. The same root is seen also in the Latin sources of our “simple” and “single.” But whether the μ of μία is related to the μ of μονάς is unclear. The latter word comes from μόνος, -η, -ον, meaning “only, alone, sole,” as we may see from English words like “monotheism.” Chantraine just traces μόνος to a conjectural *μονϜός, expressing doubt that it is related to μανός “thin, scanty.” As Smyth [, ¶e] describes it, the ending -άς, -άδος creates “ab- stract and collective numbers,” so that ἑνάς or μονάς is “the number one, unity, monad.” I think we may usefully consider the suffix as indicating

The word unité is Anglo-French, being traced to the Psautier d’Oxford of  [].   Euclid’s Foundations of Arithmetic

a set with the number of elements indicated by the main word. Thus a δεκάς, a decade, is a set of ten years. As far as I can tell though, the only specific number that Euclid uses is δύας, in ix. and later: it is a pair or double or dyad. Meanwhile, our question was whether the definition of unity now in Euclid is authentic, and in particular whether some words of Iamblichus bear on this question. Those words are from On Nicomachus’s Introduc- tion to Arithmetic [, p. , l. ]. The relevant paragraph begins as follows; as there seems to be no full published English translation, I give my own attempt. Μονὰς δέ ἐστι ποσοῦ τὸ ἐλάχιστον ἢ ποσοῦ τὸ πρῶτον καὶ κοινὸν μέρος ἢ ἀρχὴ ποσοῦ· ὡς δὲ Θυμαρίδας περαίνουσα ποσότης, ἐπεὶ ἑκάστου καὶ ἀρχὴ καὶ τέλος πέρας καλεῖται, ἔστι δὲ ὧν καὶ τὸ μέσον, ὥσπερ ἀμέλει κύκλου καὶ σφαίρας. ὁι δὲ νεώτεροι καθ´ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται· ἔλειπε δὲ τῷ ὅρῳ τούτῳ τὸ κἂν συστηματικὸν ᾖ. συγκεχυμένως δὲ οἱ Χρυσίππειοι λέγοντες ‘μονάς ἐστι πλῆθος ἕν’· μόνη γὰρ αὕτη ἀντιδιέσταλται τῷ πλήθει. // Unity is the least of an amount or the first and common part of an amount or the beginning of an amount: thus Thymaridas [called it] “[the] limiting quantity,” since the beginning and end of everything is called a limit. (But there are things of which there is a middle, such as, of course, the What does circle and sphere.)5 More recent authorities [called unity] “that ac- “abesse cording to which each entity is said to be one thing”; but they left malim” mean? out the restriction, “even though it be collective.” The Chrysippians, confusedly, saying, “Unity is multitude one”; for one is opposed to multitude.

Thymarides was “an ancient Pythagorean, probably not later than Plato’s time” [, p. ]. Since Euclid was later than Plato’s time, it is not clear whether Iamblichus’s “more recent authorities” (νεώτεροι) included

The Greek text names the work as being of Iamblichus of Chalcis in the valley of Syria (ΙΑΜΒΛΙΧΟΥ ΧΑΛΚΙ∆ΕΩΣ ΤΗΣ ΚΟΙΛΗΣ ΣΥΡΙΑΣ). The clause “even though it be collective” is Heath’s [, v. II, p. ]. The three passages italicized here are just printed in spaced-out letters in Pistelli’s text. There, as in the transcription, only the third of these passages is bounded by inverted commas. Pistelli, the editor, marks this passage, “abesse malim.” . Proportion 

Euclid or not. As suggested above, Russo [, p. ] thinks Euclid is not one of these authorities. Heath [, p. ] apparently thinks he is. At any rate, Heath continues to refer to the definition of unity in the Elements as “Euclid’s definition,” even though he has taken note of Iamblichus’s observation of what was missing from this definition. Apparently, according to Iamblichus, the full definition should be something like, Unity is that according to which each entity is said to be one thing, even though it be collective. I see no indication by Heath or Russo that such a definition can be found anywhere else than in Iamblichus’s comment. However, it does bear some resemblance to a definition that Sextus Empiricus attributes to Plato in § of “Against the Arithmeticians” (which is Book IV of Against the Professors, here from the edition of Bekker [, p. , l. ], with the translation of Bury [, pp.  f.]): Τὴν τοῦ ἑνὸς τοίνυν νόησιν διατυπῶν ἡμῖν πυθαγορικώτερον ὁ Πλάτων φησὶν ‘ἕν ἐστιν οὗ μηδὲν χωρὶς λέγεται ἕν’ ἢ ‘οὗ μετοχῇ ἕκαστον ἕν τε καὶ πολλὰ λέγεται.’ τὸ γὰρ φυτόν, εἰ τύχοι, καὶ τὸ ζῶον καὶ ὁ λίθος προσαγορεύεται μὲν ἕν, οὐκ ἔστι δὲ κατὰ τὸν ἴδιον λόγον ἕν, ἀλλ´ ἐν μετοχῇ ἑνὸς νοεῖται, τούτου μηδενὸς τούτων καθεστῶτος. // Now Plato, in formulating in rather Pythagorean fashion the concept of the one, declares that “One is that without which nothing is termed one,” or “by participation in which each thing is termed one or many.” For the plant, let us say, or the animal, or the stone is called one, yet is not one according to its own proper description, but is conceived as one by participation in the One, none of them actually being the One. This has little to do with mathematics, and neither does Sextus’s entire essay. Sextus does not give me the impression of somebody who knows what Euclid is about. It is plausible that the definition of unity now in the Elements was a later addition.

. Proportion

What we refer to as proportion could also be called analogy. The Greek for proportional is ἀνάλογον, and Euclid defines it for numbers, in what   Euclid’s Foundations of Arithmetic is now listed as the twentieth definition at the head of Book vii:

᾿Αριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ δευτέρου καὶ ὁ τρίτος τοῦ τετάρτου ἰσάκις ᾖ πολλαπλάσιος ἢ τὸ αὐτὸ μέρος ἢ τὰ αὐτὰ μέρη ὦσιν. // Numbers are proportional when the first is of the second, and the third is of the fourth, equally multiple, or the same part, or the same parts.

We saw the “definitions” of part and parts in §. (p. ). These are followed by:

Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος. // And the greater [number] is a multiple of the less when it is measured by the less.

The notions of multiple and part are thus correlative, and we have three ways to say the same thing: ) B is a multiple of A; ) A is a part of B; ) A measures B. Measurement is the basic undefined notion. Proportionality of magnitudes is defined at the head of Book v:

᾿Εν τῷ αὐτῷ λόγῳ μεγέθη λέγεται εἶναι πρῶτον πρὸς δεύτερον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ πρώτου καί τρίτου ἰσάκις πολ- λαπλάσια τῶν τοῦ δευτέρου καὶ τετάρτου ἰσάκις πολλαπλασίων καθ᾿ ὁποιονοῦν πολλαπλασιασμὸν ἑκάτερον ἑκατέρου ἢ ἅμα ὑπερέχῃ ἢ ἅμα ἴσα ᾖ ἢ ἅμα ἐλλείπῇ ληφθέντα κατάλληλα. Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον καλείσθω. // Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when equimultiples of the first and third, by whatever multiplica- tion, are respectively either alike in excess of, or alike equal to, or alike falling short of, equimultiples of the second and fourth. And magnitudes having the same ratio are called proportional.

In the definition of proportionality of numbers, Euclid does not mention ratios; but he does mention them later, as for example in Propositions  and  in Book vii: . Proportion 

᾿Εὰν ἀριθμὸς δύο ἀριθμοὺς πολλαπλασιάσας ποιῇ τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς πολλαπλασιασθεῖσιν. // If a number multiply two numbers, the numbers produced will have the same ratio as the multiplicands. A third way of referring to numbers in proportion is seen in Proposition vii.: ᾿Εαν ᾖ ὡς ὅλος πρὸς ὅλον, οὕτως ἀφαιρεθεὶς πρὸς ἀφαιρεθέντα, καὶ ὁ λοιπὸς πρὸς τὸν λοιπὸν ἔσται, ὡς ὅλος πρὸς ὅλον. // If as whole be to whole, so subtrahend to subtrahend, also remainder to remainder will be, as whole to whole.

Thus, whether we are working with arbitrary magnitudes or numbers we have different ways of expressing what seems to be the same thing. We can say in words that ) A, B, C, and D are proportional, or ) A is to B as C is to D, or ) A has the same ratio to B that C has to D. I do not see an important distinction to make between these modes of expression as such. We might abbreviate any of them by writing

A : B :: C : D. ( ) ∗ However, it is important not to write A : B = C : D. Neither definition of proportion describes an equation of two things. In a proportion, two ratios are not equal, but the same. This is related to the fact that a ratio is not a thing that can be drawn in a diagram; it is a relation between two things in a diagram—two things that might have the same relation to one another that two other things have. (See §., p. , for a possible exception to the rule that a ratio cannot be drawn.) I used the expression equally multiple in the Book-vii definition of proportion, and equimultiples in the Book-v definition; but these stand for the same Greek phrase ἰσάκις πολλαπλάσιος, which however appears in singular form in Book vii, plural in Book v. Heath uses “same multiple” for my equally multiple; but I have followed Heath in using the word “equimultiple” in the other case. Like “unit,” the word “equimultiple”   Euclid’s Foundations of Arithmetic seems to have been coined for translating Euclid: the earliest example in the Oxford English Dictionary is again from Billingsley’s  translation. In any case, Euclid’s ἰσάκις is an adverb derived from the adjective ἰσός “equal”; it is not the adjective itself. The definitions of proportionality do not describe things that are equal, though they may describe multiplying two things equally. In Greek as in English, it appears there is no adverb “samely.” Instead of ( ), much less should we write A/B = C/D, since A/B indicates a fraction,∗ and none is suggested by Euclid’s definition. Still, the important question is what we mean by writing ( ) or something like it. ∗

. Sets

In addition to distinguishing equality from identity, we should distinguish measurement from division. Given twelve apples, we can describe the same operation in two ways: we can measure the twelve apples by three apples, or we can divide them into four equal parts. Euclid does refer to dividing, as distinct from measuring, as for example in the sixth definition of Book vii: ῎Αρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος. An even number is one that is divided in twain. I use the archaic “twain” here because it is usually seen only in the phrase “in twain,” and this translates the single Greek adverb δίχα. Euclid’s numbers seem to be indistinguishable from our nonempty finite sets. After the definition of unit or unity in the Elements (quoted in §., p. ), there comes, ᾿Αριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος. And a number is a multitude of unities. In addition to the expression multitude, which Heath uses, other possible translations of πλῆθος are “mass, throng, crowd” []. We use the word “number” in this way too, as when we say a number of people are marching in the streets. . Sets 

In this sense, one is not a number. Thus we have, in the eleventh, twelfth, and thirteenth definitions at the head of Book vii: Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος. Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ. Σύν- θετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος. // A prime number is a number measured only by unity. Numbers prime to one another are numbers measured only by unity as a common measure. A composite number is a number measured by some number. Every number is measured by unity; if this were a number too, then by definition every number would be composite. Now, we did question the authenticity of the definition of unity; we may do the same for other definitions. But consider Proposition  of Book ix: ᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, οὐκ ἔσται ὡς ὁ πρῶτος πρὸς τὸν δεύτερον, οὕτως ὁ δεύτερος πρὸς ἄλλον τινά. // If two numbers be prime to one another, it will not be the case that the first is to the second as the second is to some other number. Unity is prime to every number, and unity will be to a number as that number is to its product with itself. Thus unity is not a number in the sense of Proposition ix.. The definitions of prime and composite numbers are imprecise: else- where, a number is allowed to measure itself. For example, Proposition vii. is the problem of finding the greatest common measure of two num- bers that are not prime to one another. If one of the numbers measures the other, then it is the greatest common measure, since it also measures itself. Strictly then, a composite number is a number measured by some number, a number of times. Throughout the number-theoretic books of the Elements, numbers are diagrammed as line segments, which are not obviously sets of units. We shall consider this feature of the Elements in §.; for now I note a curious move that Euclid makes in proving Proposition vii.. The enunciation is,

In Heiberg’s Greek text, they are th, th, and th; but Heiberg brackets Defi- nition  and omits it from his Latin translation, renumbering the later definitions accordingly. Heath follows suit.   Euclid’s Foundations of Arithmetic

᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, ἅπερ ἀφαιρεθεὶς ἀφαιρεθέντος, καὶ ὁ λοιπὸς τοῦ λοιποῦ τὰ αὐτὰ μέρη ἔσται, ἅπερ ὁ ὅλος τοῦ ὅλου. // If a number be the very parts of a number that a subtrahend is of a subtrahend,7 also the remainder will be the very same parts of the remainder that the whole is of the whole.

In our symbols, if A : B :: C : D, then A C : B D :: A : B. In Euclid’s proof, the number ΑΒ is assumed to be− the same− parts of Γ∆ that the subtrahend ΑΕ is of the subtrahend ΓΖ. These numbers are drawn out as line segments, with Ε lying on ΑΒ, and Ζ on Γ∆. We want to break up ΑΒ into parts, each equal to the same part of Γ∆; and we want to break up ΑΕ into the equally numerous parts of ΓΖ. Doing this directly makes the diagram complicated; so Euclid takes ΗΘ equal to ΑΒ and divides it instead. Here ΗΘ is different from ΑΒ as a set of units; but the two sets are equipollent, or as Euclid says, equal. The above definition of evenness of a number is meaningful for an arbitrary set, possibly infinite: a set B is even, just in case it can be divided in two, in the sense of having a subset A for which there is a bijection f from A to B r A. Then the collection  x,f(x) : x A partitions B into two-element subsets: this means B{is measured} ∈by a} two-element set, a dyad. Conversely, being measurable by a dyad implies being even, but not in bare Zermelo–Fraenkel set theory; a choice principle is needed, albeit a principle that is weaker than the full Axiom of Choice [, . & .]. Thus there is mathematical reason to distinguish between measuring by two and dividing in two, as Euclid’s language does.

. Parts

If A, B, C, and D are numbers, or possibly units, and A measures B, then presumably ( ) (on p. ) holds if and only if C measures D and ∗ Heath uses “number subtracted” for my subtrahend; but I remember being taught the latter word in third grade, along with its correlate, “minuend.” According to the OED, the words go back three centuries in English. . Parts  measures it the same number of times—a situation we might describe by saying that for some multiplier n,

nA = B & nC = D.

In this case, also B : A :: D : C. In Definition vii. then, it remains to understand what being “the same parts” means. By Definition vii., if A is less than B, but does not measure B, then A is not part of B, but is “parts” of B. Heath’s entire comment on the definition of “parts” is the following.

By the expression parts (μέρη, the plural of μέρος) Euclid denotes what we should call a proper fraction. That is, a part being a submultiple, the rather inconvenient term parts means any number of such submultiples making up a fraction less than unity. I have not found the word used in this special sense elsewhere, e.g. in Nicomachus, Theon of Smyrna or Iamblichus, except in one place of Theon (p. , ) where it is used 2 of a proper fraction, of which 3 is an illustration. This note is misguided in two ways. It ignores the fact that “parts” is defined by simple negation, as when we define an irrational number to be a real number that is not rational. We might then say that an irrational number is one whose decimal expansion is neither finite nor repeating; but this would have to be proved. In commenting on the definition of “parts,” Heath is probably thinking ahead to Proposition vii., whose enunciation we saw in §. (p. ): “Every number is of every number, the less of the greater, either a part or parts.” The demonstration of this proposition is not a simple appeal to the definition of “parts.” Thus that definition at the head of Book vii is not really a definition, but is a kind of summary of what is to come in the book. We shall consider the demonstration of Proposition vii. in §. (page ) below. Meanwhile, let us note that, pace Heath, the proposition is not likely to be about fractions. According to David Fowler [, §.(b), pp. –],

We have no evidence for any conception of common fractions p/q and their manipulations such as, for example, p/q r/s = pr/qs and × p/q + r/s = (ps + qr)/qs, in Greek mathematical, scientific, financial,   Euclid’s Foundations of Arithmetic

or pedagogical texts before the time of Heron and Diophantus; and even the fractional notations and manipulations found in the Byzan- tine manuscripts of these late authors may have been revised and in- troduced during the medieval modernization of their minuscule script. Among the thousands, possibly the tens of thousands, of examples of fractions to be found in contemporary Egyptian (hieroglyphic, hieratic, and demotic), Greek, and Coptic texts, all but a few isolated examples in five texts . . . use throughout the following ‘Egyptian[’] system for expressing fractions . . .  We take the basic sequence of the arithmoi:

two, three, four, five, . . . ,

represented in Greek by the letters β, γ,δ,ǫ,... , and convert it to the sequence

half, third, quarter, fifth, . . . ,

where, after the exceptional cases of the first few terms, for which special symbols . . . are assigned, the derived symbol is . . . usually ′ ′ ′ ′ ′ ′ transcribed as an accent, γ, δ, ǫ,... , or a prime, γ ,δ ,ǫ ,... ′ The sequence of parts starts with β, ‘the two parts’, τὰ δύο μέρη, an expression for ‘two-thirds’. However extraordinary it may seem to us, it is an incontrovertible fact that the sequence of parts starts with two-thirds . . . (This is an additional reason for avoiding the name ‘unit ′ ′ ′ fractions’ for the sequence β, ∠, γ, δ,... )... More complicated fractions than simple parts are expressed as sums of an integer and different simple parts . . . Fowler goes on to describe “division tables” [, §.(a), pp.  f.]:

12 our common fraction 17 . . . would be expressed in a division table . . . as

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ τῶν ιβ [τὸ ιζ] ∠ιβιζλδναξη ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ of the 12 [the 17th is] 21217345168

Widely spaced dots “ . . . ” are my ellipses; narrowly spaced dots “. . . ” denoting continuations of sequences are Fowler’s. . Music 

for what we would write as

12 1 1 1 1 1 1 17 = 2 + 12 + 17 + 34 + 51 + 68 .

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ I think a better “translation” of the Greek ∠ιβιζλδναξη would be some- thing like ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ 2 10 2 10 7 30 4 50 1 60 8. Possibly the definition of “parts” in the Elements serves as a reminder to the reader that, even though  is not a part of , it is still parts of , namely the half, and the twelfth, and the seventeenth, and the thirty- fourth, and the fifty-first, and the sixty-eighth. In this case, Euclid’s Proposition vii. will give a new meaning to the notion of “parts.”

. Music

In the diagrams of the propositions in Books vii–ix, numbers are bounded straight lines (what are also called today line segments). Heath remarks on this after Proposition vii.: the representation in Books vii. to ix. of numbers by straight lines is adopted by Heiberg from the mss. The method of those editors who substitute points for lines is open to objection because it practically ne- cessitates, in many cases, the use of specific numbers, which is contrary to Euclid’s manner. Heath is right that points should not be used, but if I understand him correctly, he is wrong about the reason. I think he means that Euclid’s propositions are general, but that this would be belied by diagrams that indicated specific numbers. However, as they are, Euclid’s proofs are of- ten not general in form, regardless of the diagram. For example, in the proof of Proposition vii., showing one use of what we call the Euclidean algorithm, there are three alternations of subtractions, not some unspec- ified number of them. One must simply understand that there is nothing

I note the oddity of English in using the exceptional form “first” here, rather than “oneth.”   Euclid’s Foundations of Arithmetic special about three alternations as being three. Likewise, throughout Book i of the Elements, specific triangles in the diagrams are to be un- derstood as general. Specific numbers of dots in a diagram could be understood as general. However, if unities could always be thought of as points, then it would be obvious that multiplication was commutative. Four rows of dots, of three dots each, obviously have as many dots as three rows, of four dots each (Figure .). With diagrams of dots, there would be no need to

b b b b b b b b b b b b b b b b b b b b b b b b

Figure .. Multiplication of points prove commutativity of multiplication, which is Proposition vii.. But it is not obvious that four straight lines, of  units each, have together the same length as three straight lines, of  units each (Figure .).

b| | b | | b | | b | | b

b| | | b | | | b | | | b

Figure .. Multiplication of straight lines

Another possible reason why Euclid’s numbers are straight lines is that lyre strings are like straight lines, and Euclid’s study of numbers is in- spired by music. According to Andrew Barker in Greek Musical Writings [, p. ], the ancient work called Sectio Canonis (Κατατομὴ κανόνος) is

attributed to Euclid in the manuscripts, and by Porphyry, who quotes from it at length . . . Parts of the treatise are also quoted by Boethius . . . [T]he attribution has been debated . . . There are no good reasons, how- ever, for denying Euclid’s authorship of the main part of the treatise, at least as much as Porphyry quotes . . . . Music 

Fowler is more skeptical. He discusses the Sectio Canonis thoroughly [, pp. –]; I would just note that the chief burden of the treatise seems to be the following. An interval of musical notes somehow corresponds to a ratio of numbers, and those ratios are of three possible kinds, as in the Elements. Indeed, according to the Sectio Canonis [, p. ], πάντα δὲ τὰ ἐκ μορίων συγκείμενα ἀριθμοῦ λόγῳ λέγεται πρὸς ἄλληλα, ὥστε καὶ τοὺς φθόγγους ἀναγκαῖον ἐν ἀριθμοῦ λόγῳ λέγεσθαι πρὸς ἀλλήλους· τῶν δὲ ἀριθμῶν οἱ μὲν ἐν πολλαπλασίῳ λόγῳ λέγονται, οἱ δὲ ἐν ἐπιμορίῳ, οἱ δὲ ἐν ἐπιμερεῖ, ὥστε καὶ τοὺς φθόγγους ἀναγκαῖον ἐν τοῖς τοιούτοις λόγοις λέγεσθαι πρὸς ἀλλήλους. // All things composed of parts are said to be to one another in the ratio of a number [to a number]; so notes too must be said to be to one another in the ratio of a number. Some numbers are said to be in multiple ratio, others in part-again, still others in parts-again [ratio]; so notes too must be said to be in in these ratios to one another. Numbers A and B are in ) multiple ratio, if A is a multiple of B; ) part-again ratio, if A B measures B, so that A is the whole of B with a part added; − ) parts-again ratio, otherwise. The latter two kinds of ratio—ἐπιμόριος and ἐπιμερής—are also called in English “superparticular” and “superpartient,” as in D’Ooge’s translation of Nicomachus [, pp. –, , & ]; but Barker uses the words “epimoric” and “epimeric” [, p. , p.  n. ]. The quoted passage from the Sectio Canonis is given in the big Liddell–Scott–Jones lexicon [] as an example of the use of ἐπιμόριος. This word is derived from μόριον rather than μέρος; but either of these means “part.” The passage continues somewhat obscurely. τούτων δὲ οἱ μὲν πολλαπλάσιοι καὶ ἐπιμόριοι ἑνὶ ὀμόματι λέγονται πρὸς ἀλλήλους. // Of these, the multiple [ratios] and the part-again [ratios] to one another are said with a single name. Barker’s translation is, “of these, the multiple and the epimoric are spoken of in relation to one another under a single name”; Fowler changes “in relation” to “with respect.” Barker suggests that “single name” refers to   Euclid’s Foundations of Arithmetic

A B C D E F G H

Figure .. Lyre strings as numbers the single words like τριπλάσιος (“triple”) and ἐπίτριτος (“third-again”) that name specific multiple and part-again ratios. Fowler goes further, suggesting that “single name” alludes to the single number that specifies one of these ratios, as the number three specifies the triple and third-again ratios. Fowler’s suggestion would seem to be partially corroborated by the last three propositions of Book vii of the Elements, which we shall consider at the end, in §. (p. ). Meanwhile, suppose the strings of an eight-stringed lyre are repre- sented by the numbers A through H, as in Figure .. As I understand Barker in The Science of Harmonics in Classical Greece [, p. ], we can consider A, D, E, and H as fixed, while the other strings can vary ac- cording to the chosen musical mode. The interval AD or EH is a fourth or diatesseron (διὰ τεσσάρων), spanning, as it does, four strings; AE or EH is a fifth or diapente (διὰ πέντε); AH, an octave or diapason (διὰ πασῶν). These intervals are all concords, and the octave is composed of a fourth and a fifth. The double octave is also a concord, but the double fourth and the double fifth are not. So much is observed or at least assumed. Then we propose the axiom that a concord corresponds to a ratio of numbers that is either multi-

Apparently such lyres existed, although the traditional lyre had seven strings [, pp.  & ]. . The Euclidean Algorithm  ple or part-again. As the commentators point out, this axiom does not correspond exactly to the observed reality. Nonetheless, be he Euclid or somebody else, the author of the Sectio Canonis uses the axiom to argue as follows. Since the double octave has a mean (namely the oc- tave), it cannot be part-again; so it must be multiple, and therefore the octave itself must be multiple. Similarly, the fifth and the fourth cannot be multiple, so they are part-again. The least multiple ratio, namely the double, is composed of the two greatest part-again ratios, namely the half- again (ἡμιόλιος, half-whole or “hemiolic”) and the third-again (ἐπίτριτος, “epitritic”). Therefore these three ratios must correspond respectively to the octave, the fifth, and the fourth. Again, it may be questioned whether there is a purely mathematical proof of this correspondence between musical intervals and numerical ra- tios. The point for now is the importance given to two kinds of numerical ratios: the ratio that obtains between the greater A and the less B when B measures A, or when B is almost equal to A, but the difference mea- sures B. These are situations when the Euclidean Algorithm concludes in one or two steps.

. The Euclidean Algorithm

The first proposition of Book vii of the Elements is a theorem in the sense of Pappus (page ); the next two are problems. All three involve the so-called Euclidean Algorithm. The enunciation of the theorem is,

∆ύο ἀριθμῶν ἀνίσων ἐκκειμένων, ἀνθυφαιρουμένου δὲ ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, ἐὰν ὁ λειπόμενος μηδέποτε καταμετρῇ τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ μονάς, οἱ ἐξ ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλὴλους ἔσονται. // Two numbers being given, and the less being alternately subtracted from the greater continually, if the remainder never mea- sures the previous number until unity is left, the original numbers will be prime to one another.

In the Greek, the phrase being alternately subtracted is one word, which is a passive participle of the verb ἀνθυφαιρέ-ω. This is the source of the   Euclid’s Foundations of Arithmetic

Ε

Γ Η ∆

Θ Α Ζ Β

Figure .. Proposition vii. feminine noun ἀνθυφαίρεσις, which I shall render as anthyphaeresis. In the exposition of the proposition, unequal numbers ΑΒ and Γ∆ are given, as in Figure ., which is Heiberg’s diagram rotated counterclock- wise through a right angle. In the demonstration, which depends on the diagram,

ὁ μὲν Γ∆ τὸν ΒΖ μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν ΖΑ, ὁ δὲ ΑΖ τὸν ∆Η μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν ΗΓ, ὁ δὲ ΗΓ τὸν ΖΘ μετρῶν λειπέτω μονάδα τὴν ΘΑ. // Let Γ∆, measuring ΒΖ, leave less than itself, ΖΑ; then let ΑΖ, measuring ∆Η, leave less than itself, ΗΓ; then let ΗΓ, measuring ΖΘ, leave a unit, ΘΑ.

It is not mentioned how many times Γ∆ measures ΒΖ; or ΑΖ, ∆Η; or ΗΓ, ΖΘ; but by definition, the unit ΘΑ measures ΗΓ, that many times. The four numbers of times here form a sequence, [n0, n1, n2, n3], which we we may call the anthyphaeretic sequence of ΑΒ and Γ∆. Fowler calls the anthyphaeretic sequence a “ratio,” or more precisely “anthyphairetic ratio” [, §., p. , & §., p. ]. He does this, because by one possible definition, A, B, C, and D are proportional just in case the pairs (A, B) and (C,D) have the same anthyphaeretic sequence. It does not matter whether A, B, C, and D are numbers or

Other writers use the transliteration “anthyphairesis,” but I follow the tradition of writing the Latin diphthong ae for the Greek diphthong αι. Presumably one will pronounce the word as something like “An thigh FEAR iss iss.” . The Euclidean Algorithm  arbitary magnitudes. In modern terms, if we compute

1 a n0 + = , 1 b n1 + 1 n2 + n3 then the ratio of ΑΒ to Γ∆ above is that of a to b. Moreover, a and b will automatically be relatively prime (assuming we compute them in the obvious way, replacing n2 + 1/n3 with (n2n3 + 1)/n3 and so forth). For arbitary magnitudes, the anthyphaeretic sequence might be infinite; however, if the magnitudes are lengths constructible with straightedge and compass, then the anthyphaeretic sequence will be periodic. The historical situation is laid out by Thomas at the end of the first of the two Loeb Classical Library volumes, Selections Illustrating the History of Greek Mathematics [, pp. –]. According to the first proposition of Book vi of the Elements, Τὰ τρίγωνα καὶ τὰ παραλληλόγραμμα τὰ ὑπὸ τὸ αὐτὸ ὕψος ὄντα πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις. // Triangles and parallelograms that are under the same height are to one another as their bases. In the Topics, Aristotle uses this result (or at least the part about par- allelograms) as an example of something that is immediately clear, once one has the correct definition; and the definition of “same ratio” is “hav- ing the same antanaeresis (ἀνταναίρεσις).” In a comment on the pas- sage, Alexander of Aphrodisias observes that Aristotle uses the word “antanaeresis” for anthyphaeresis. In , Oskar Becker observed that Aristotle and Alexander could be alluding to the anthyphaeretic defi- nition of proportion described above. It is thus reasonable to suppose that Euclid was aware of the possibility of an anthyphaeretic theory of proportion. Proposition vii. is a problem: ∆ύο ἀριθμῶν δοθέντων μὴ πρώτων πρὸς ἀλλήλους τὸ μέγιστον αὐτῶν κοινὸν μέτρον εὑρεῖν. // Two numbers not prime to one another being given, to find their greatest common measure.   Euclid’s Foundations of Arithmetic

We may ask why this proposition is separated from Proposition , since each of these propositions involves an application of the Euclidean Al- gorithm, that is, anthyphaeresis. But Proposition  itself considers two cases: when the the less of given numbers measures the greater, and when it does not. We see here the practice of using language as precisely as possible—a practice often abjured today in mathematics, in favor of simplicity. If a straight line is erected on another straight line, forming two angles, today we should say that the sum of the angles is two right angles; for Euclid, in Proposition i., either this is so, or each of the an- gles is already a right angle. Today we have an algorithm, the Euclidean Algorithm, for finding greatest common divisors; Euclid has an algorithm for verifying that two numbers are relatively prime, and for finding their greatest common measure when they are not. There is now a break in the usual pattern of exposition. Proposition vii. has a πορίσμα, a porism: in the words of Proclus, “a kind of windfall I think we are or bonus in the investigation” [, p. ], namely: not going to need this ᾿Εκ δὴ τούτου φανερόν, ὅτι ἐὰν ἀριθμὸς δύο ἀριθμοὺς μετρῇ, καὶ τὸ porism μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει. // From this then it is clear that, if a number measure two numbers, it will also measure their greatest common measure.

As a corollary follows from the enunciation of a theorem, a porism follows from the proof. The term could be used more widely today, as for example to label a generalization of a theorem, when it turns out that the same proof yields the generalization. Proposition vii. is to find the greatest common measure of three given numbers. There seems to be no reason to look further at this before moving on to Proposition .

. Part or Parts

We have seen the enunciation of Proposition vii. in §§. & . (pp.  & ); it is that the less number is part or parts of the greater number. If being parts of a number just meant not being part of the number, that is, . Part or Parts  not measuring the number, then Proposition  would be immediate. Eu- clid does not treat it as immediate, and so the real meaning of parts must be more subtle. We should consider what Euclid actually says. After the enunciation of Proposition  come the exposition and specification: ῎Εστωσαν δύο ἀριθμοὶ οἱ Α, ΒΓ, καὶ ἔστω ἐλάσσων ὁ ΒΓ· λέγω, ὅτι ὁ ΒΓ τοῦ Α ἤτοι μέρος ἐστὶν ἢ μέρη. // Let the two numbers be Α and ΒΓ, and let the less be ΒΓ. I say that ΒΓ is, of Α, either part or parts. The demonstration then considers two cases: Οἱ Α, ΒΓ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. // For Α and ΒΓ are either prime to one another, or not. The two cases thus correspond to Propositions vii. and vii.. Here is the first case: ἔστωσαν πρότερον οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους. διαιρεθέντος δὴ τοῦ ΒΓ εἰς τὰς ἐν αὐτῷ μονάδας ἔσται ἑκάστη μονὰς τῶν ἐν τῷ ΒΓ μέρος τι τοῦ Α· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α. // First suppose Α and ΒΓ are prime to one another. ΒΓ being divided into the units in itself, each unit of those that are in ΒΓ will be some part of Α. Thus ΒΓ is parts of Α. The word thus here translates Euclid’s ὥστε. As “thus” can mean either therefore or in this way, so can ὥστε. I propose that in the present instance it has the latter meaning. That is, since Α and ΒΓ are coprime, ΒΓ is parts of Α, in the sense that each of its units is a part of Α. Sometimes, indeed, ὥστε simply means therefore, only less emphati- cally; Heath translates it as “so that.” Euclid’s usual word to indicate a logical conclusion is ἄρα, used postpositively—not at the beginning of a clause; but occasionally ὥστε is used in place of this, only preposi- tively—at the beginning. For example, Proposition vii. is to find the least common multiple of two numbers, and Proposition vii. is that this least common multiple measures every common multiple. Proposi- tion vii. is to find the least common multiple of three numbers, given as Α, Β, and Γ. One lets ∆ be the least common multiple of Α and Β. In case Γ does not measure this, one lets Ε be the least common multiple of Γ and ∆, so that Ε is a common multiple of Α, Β, and Γ. If it is not the least such, one lets Ζ be less. Euclid’s argument continues as follows.   Euclid’s Foundations of Arithmetic

ἐπεὶ οἱ Α, Β, Γ τὸν Ζ μετροῦσιν, καὶ οἱ Α, Β ἄρα τὸν Ζ μετροῦσιν· καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Α, Β μετρούμενος τὸν Ζ μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν Α, Β μετρούμενός ἐστιν ὁ ∆· ὁ ∆ ἄρα τὸν Ζ μετρεῖ. μετρεῖ δὲ καὶ ὁ Γ τὸν Ζ· οἱ ∆, Γ ἄρα τὸν Ζ μετροῦσιν· ὥστε καὶ ὁ ἐλάχιστος ὑπὸ τῶν ∆, Γ μετρούμενος τὸν Ζ μετρήσει. // Since Α, Β, and Γ measure Ζ, therefore also Α and Β measure Ζ; therefore also the least [number] measured by Α and Β will measure Ζ. But the least [number] measured by Α and Β is ∆. Therefore ∆ measures Ζ. But also Γ measures Ζ; therefore ∆ and Γ measure Ζ; thus also the least [number] measured by ∆ and Γ will measure Ζ. The last ὥστε replaces ἄρα, perhaps because a pattern has already been established. Then ὥστε indicates an easy result. This is seen also in Book i, where ὥστε is used three times, in Propositions , , and , to indicate a result of the form

A = C and B = C, thus A = B; and in Propositions , , and , for similarly easy results. But in Proposition i., which we looked at in §., ὥστε is used to indicate the conclusions that Γ falls on Ζ, and ΒΓ on ΕΖ, and triangle ΑΒΓ on ∆ΕΖ; and in Proposition i., ὥστε indicates again that ΑΒΓ coincides with ∆ΕΖ. In short, ὥστε indicates that something happens, thus. We see ὥστε again in the second case of Proposition vii., or at least in its second subcase (corresponding to the second case of Proposition vii.): Μὴ ἔστωσαν δὴ οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους· ὁ δὴ ΒΓ τὸν Α ἤτοι μετρεῖ ἢ οὐ μετρεῖ. // Now suppose Α and ΒΓ are not prime to one another. Then ΒΓ either measures Α or does not measure. εἰ μὲν οὖν ὁ ΒΓ τὸν Α μετρεῖ, μέρος ἐστὶν ὁ ΒΓ τοῦ Α. // If ΒΓ measures Α, ΒΓ is a part of Α. The second subcase is diagrammed in Figure .. εἰ δὲ οὔ, εἰλήφθω τῶν Α, ΒΓ μέγιστον κοινὸν μέτρον ὁ ∆, καὶ διῃρήσθω ὁ ΒΓ εἰς τοὺς τῷ ∆ ἴσους τοὺς ΒΕ, ΕΖ, ΖΓ. καὶ ἐπεὶ ὁ ∆ τὸν Α μετρεῖ, μέρος ἐστὶν ὁ ∆ τοῦ Α· ἴσος δὲ ὁ ∆ ἑκάστῳ τῶν ΒΕ, ΕΖ, ΖΓ· καὶ . Part or Parts 

∆

Β Ε Ζ Γ

Α

Figure .. Proposition vii.

ἕκαστος ἄρα τῶν ΒΕ, ΕΖ, ΖΓ τοῦ Α μέρος ἐστίν· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α. // If not, let the greatest common measure ∆ of Α and ΒΓ be taken, and let ΒΓ be divided into ΒΕ, ΕΖ, and ΖΓ, equal to ∆. Since ∆ measures Α, ∆ is a part of Α. And ∆ is equal to each of ΒΕ, ΕΖ, and ΖΓ. Therefore each of ΒΕ, ΕΖ, and ΖΓ is a part of Α; thus ΒΓ is parts of Α. The conclusion of the proposition repeats the enunciation, with—as is usual—the addition of ἄρα and a qed: ῞Απας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη· ὅπερ ἔδει δεῖξαι. // Therefore every number is of every number, the less of the greater, either a part or parts; which is just what was to be shown. The last verb, δεῖξαι, is the normal ending for a theorem, as opposed to a problem, which would end with ποιῆσαι “to be done.” Nonetheless, strictly speaking, Proposition vii. is neither a theorem nor a problem in the usual sense, because its enunciation alone gives us nothing that we can either contemplate or use. The whole proposition is an explanation of the definition of proportion. It tacitly explains how to tell when a number A is the same parts of B that C is of D. Assuming that A is not in fact part of B, and C is not part of D, we may conclude that A is the same parts of B that C is of D, just in case A and B are, respectively, the same multiples of their greatest common measure that C and D are of their greatest common measure. If Euclid spelled out this definition, he might distinguish the case where, say, A and B are prime to one another. In this case, A is the   Euclid’s Foundations of Arithmetic

∆ Γ Γ

∆

Α Β Α Β

Figure .. Proposition i., with the missing case same parts of B that C is of D, just in case C = A E and D = B E, where E is the greatest common measure of C and×D. × Euclid does not spell out the details of the definition of having the same parts. This need not be considered unusual. In Book i of the Elements, there are several propositions where a thorough account of all possible cases is lacking. Euclid does not even bother to mention that there are other cases than the one he considers. We have no reason to think that he is not aware of the other cases. For example, Proposition i. is that, in an isosceles triangle, not only () the base angles, but also () the exterior angles at the base, are equal to one another. A consequence of this is Proposition , which is that two different points on the same side of a straight line cannot have the same respective distances from its endpoints. In his proof by contradiction, Euclid considers only the case shown on the left of Figure ., where neither of the two points Γ and ∆ lies inside the triangle determined by the other point and the given straight line ΑΒ. The proof of this case needs only the first conclusion of Proposition . The other case goes unmentioned, but is proved by means of the second conclusion of Proposition . As discussed in §., by definition, a number A is to B as C is to D, provided A is () the same multiple or () part or () parts of B that C is of D. Symbolically, this means, ) for some multiplier n, both A = nB and C = nD, or ) for some multiplier n, both nA = B and nC = D, or . Part or Parts 

) for some multipliers k and ℓ, and some numbers E and F ,

A = kE, B = ℓE, C = kF, D = ℓF,

where

E = gcm(A, B), F = gcm(C,D) (†)

(where gcm means greatest common measure). Condition (†) is made as explicit as need be by Proposition . Pengelley and Richman [, p. ] leave out Condition (†) when they formulate Euclid’s definition of proportion. Then they observe that later proofs make use of the transitivity of “equality” of ratios without having proved it. However, the transitivity of this “equality” is immediate from the definition of proportion, properly understood. By definition, A is to B as C is to D, provided that something about pairs (X,Y ) of numbers is the same, whether the pair be (A, B) or (C,D). That “something” cannot be mere existence of a number Z and multipliers k and ℓ such that

X = kZ, Y = ℓZ. ( ) ‡ For, such Z, k, and ℓ exist for all pairs of numbers X and Y . The desired “something” could however be the set of all pairs (k,ℓ) of multipliers such that, for some number Z, ( ) holds. For Euclid however, as he shows in Proposition vii., the “something”‡ is the one pair (k,ℓ) such that

X = k gcm(X,Y ), Y = ℓ gcm(X,Y ). · · To a modern reader accustomed to manipulating and reducing fractions, Euclid’s meaning may not be entirely clear. But I think it is clear enough to his intended audience. Euclid is aware that proportionality—sameness of ratio—can be de- fined by anthyphaeresis. He prefers to avoid this, perhaps in order to make proofs easier. For the arbitrary magnitudes considered in Book v, he uses the definition quoted in §.. Looking at arbitrary equimultiples is conceptually easier than looking at the potentially infinite sequences   Euclid’s Foundations of Arithmetic of numbers generated by anthyphaeresis. For one thing, even if we say A measures B, k times, with remainder C, Euclid has no notation for the multiplier k. Indeed, k here is not a thing. It is not, strictly speaking, a number, but a numeral; not a noun, but an adjective. It can be turned into a noun, as it will be in Book vii, when multiplication of numbers is defined; but there is no need to do this in Book v. When anthyphaeresis is applied to numbers, finite sequences are gen- erated. We apply anthyphaeresis to numbers A and B, we get a greatest common measure C. Then A = kC and B = ℓC for some k and ℓ, as Euclid knows. He does not say much about k and ℓ directly though. He does not say, for example, that they are relatively prime. He does not have good notation for them. We can write multipliers with minuscules, for example; Euclid cannot. Now, good notation is not like house-building supplies. You cannot will bricks and mortar into existence; but if you find your notation inadequate, you just improve it. If Euclid was inhibited by a lack of good notation, it would seem the fault was his own. On the other hand, Euclid’s concern was not written expression as such. Writing supplies are like house-building supplies. Even if Euclid had as much scratch paper (or papyrus) to work with as we do, he could not easily have distributed a copy of his lectures to everybody who attended. I think of Phaedrus, in the eponymous dialogue of Plato, borrowing the text of a speech of Lysias, not so that he can copy it, but so that he can memorize it [, b, p. ]. Evidently Euclid wrote things out, and that is why we have his work; but his aim was not to come up with correct written expressions as such. See however §..

. Multiplication

In the Elements, addition of numbers is implicitly understood to be what we call a commutative, associative operation. Thus numbers compose an additive semigroup. Each number is a sum or combination of units, all of these units being equal to one another. Multiplication is explicitly defined. The fifteenth of the definitions at . Multiplication  the head of Book vii reads, ᾿Αριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις. // A number is said to multiply a number when, however many units are in it, so many times is the multiplied number composed, and some number comes to be.

I use the verb to be composed here for Euclid’s συντίθημι, because from Parse Euclid’s this verb is derived the adjective σύνθετος, and we translate this (as in form Definition vii.) as composite. Heath uses “to be added to itself” for Euclid’s συντίθημι; but taken literally, this is misleading. To multiply B Reference by A means to lay down a copy of B for each unit in A; it does not mean to... to add all of those copies to the B that already exists. If we multiply B somebody? by A—or as Euclid says, if A multiplies B—, we can describe the result as “B, A times,” or “A times B.” We can write the result as A B, × although Euclid has no such notation. As we noted in §. (p. ), it is not obvious that A B = B A, and this will actually be the result of Proposition . Nonetheless,× × the definition of multiplication is followed by: ῞Οταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί. // And whenever two numbers, multiplying one another, make some [number], the [number] produced is called plane, and its sides are the numbers multiplying one another. This suggests an understanding that multiplication is indeed commuta- tive. This contributes to the feeling that the “definitions” at the head of Book vii are more of a summary or introduction than a list of formal definitions. Given numbers A and B, we can write A B = nB, ( ) × § where n is the number or rather numeral of units in A. Here A is a noun, but n is an adjective. We can draw A in a diagram, but not n itself.   Euclid’s Foundations of Arithmetic

Rather, n would be a feature of the diagram. Instead of ( ), we could write § A B = B + + B. × ··· Another way to say this is that A B belongs to the semigroup generated by B. × Euclid distinguishes between products A B and multiples nB in his propositions. The enunciation of Proposition×  is:

᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ αὐτὸ μέρος ᾖ, καὶ συναμφότερος συναμφοτέρου τὸ αὐτὸ μέρος ἔσται, ὅπερ ὁ εἷς τοῦ ἑνός. // If a number be part of a number, and another be the same part of another, then the combination will be the very same part of the combination that the one is of the one.

Heath has “sum” instead of my combination; but the latter seems closer to Euclid’s συναμφότερος, both in meaning and etymology: the Latin com- (for con-) corresponds to συν-, while the B of bin- seems to be related to the Φ in αμφ-. If the numbers in question are respectively A, B, C, and D, we assume that, for some n, both nA = B and nC = D; the conclusion is that n(A + C)= B + D. In one line then,

n(A + C)= nA + nC; this can be understood as a consequence of the commutativity of addition. Proposition  is the same as , but with “part” replaced by “parts.” If A = kE, B = nE, C = kF , and D = nF , then A + C = k(E + F ), while B + D = n(E + F ). There is no need here to assume that k and n are relatively prime, or equivalently that E is the greatest common measure of A and B. Nonetheless, from Propositions  and , Euclid immediately obtains Proposition :

A : B :: C : D = A : B :: A + C : B + D, ⇒ or more generally

A1 : B1 :: :: An : Bn = A1 : B1 :: A1 + + An : B1 + + Bn. ··· ⇒ ··· ··· . Multiplication 

Note that this hardly needs proof, if one understands and accepts the anthyphaeretic definition of proportion. Meanwhile, Euclid has already used the idea of the general form of Proposition  in Proposition : ᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ αὐτὸ μέρος ᾖ, καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ἢ μέρη ὁ πρῶτος τοῦ τρίτου, τὸ αὐτὸ μέρος ἔσται ἢ τὰ αὐτὰ μέρη καὶ ὁ δεύτερος τοῦ τετάρτου. // If a number be a part of a number, and another be the same part of another, also alternately, what part or parts the first is of the third, the same part or parts will also the second be of the fourth. Thus if the four numbers are A, B, C, and D, and nA = B and nC = D, then whatever part or parts A is of C, the combination B will be the same part or parts of the combination D, that is, A : C :: B : D. ( ) ¶ Without specifying any hypotheses, we could write this as A : C :: nA : nC. ( ) k The possibility that A is a multiple of C is ignored, presumably because, without loss of generality, we may assume A

Unlike Proposition , this is not immediate from the anthyphaeretic definition of proportion. But we shall need it to establish commutativity of multiplication. To this end, there is Proposition , which is nearly the same as a special case of , but in the language of proportion and products, instead of parts: 1: B :: A : A B. × Then also 1 : A : B : B A, so by alternation 1 : B : A : B A; and therefore × × A B = B A. × × This is Proposition , whose enunciation is Εὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται. If two numbers, multiplying one another, make some [number], the products will be equal to ane another. This would indeed appear to be a “real” theorem. We can understand Euclid’s situation as follows. His numbers compose a set S that is equipped with () a commutative, associative operation of addition, which we denote by +; () an associative multiplication, , which distributes over addition; () a multiplicative identity, 1; ()× a linear ordering, <, such that A

Textbook references are Lam [, Exercise ., p. ] and Botto Mura and Rhem- tulla [, Thm .., pp.  f.]. Every free product of ordered groups is orderable. . Multiplication 

To apply the Euclidean Algorithm, Euclid tacitly assumes what we call the Well Ordering Property of S . I think Euclid also tacitly assumes that A B always belongs to the semigroup generated by B. Either of these assumptions× implies the other, although Euclid does not prove this or perhaps even contemplate a proof. Nonetheless, Euclid does not assume commutativity of multiplication. He proves it. This would seem to be more than number-theory text- books do today. Landau [] and Hardy & Wright [] do not discuss foundations, but start right in, proving theorems about divisibility. I suppose they expect the reader to understand the natural numbers and all of the integers as lying among the so-called real numbers, whose alge- braic properties, at least, are familiar. As far as the integers themselves are concerned, the authors evidently take for granted the axioms that LeVeque [, pp. –] makes explicit: in effect, the integers compose an ordered commutative ring whose positive elements are well-ordered. LeVeque has an additional axiom, that c = 0 and ca = cb imply a = b, but this is true in every ordered ring. 6 Leveque observes that the well-ordering axiom can be replaced with the induction axiom, once the axioms of a commutative ordered ring are assumed. Burton [, pp. –] proves the “Archimedean property” and the “First Principle of Finite Induction,” using the “Well-Ordering Principle” as the only explicit property of the natural numbers; otherwise, he says,

We shall make no attempt to construct the integers axiomatically, as- suming instead that they are already given and that any reader of this book is familiar with many elementary facts about them.

It is presently asserted [, p. ] that “Mathematical induction is often used as a method of definition as well as a method of proof.” This sug- gests that recursive definitions are justified by induction alone. They are not, and clarifying this point is of value to number theory itself, as we discussed above (page ). Again, in our terms, Euclid proves that every ordered ring whose posi- tive elements are well-ordered is commutative. Thus he would appear to be more sensitive to rigor and logical economy than are the corresponding texts of today.   Euclid’s Foundations of Arithmetic

. Euclid’s Lemma

“Euclid’s Lemma” now follows from some straightforward manipulations. Thus Proposition vii. is

C : D :: A C : A D, ( ) × × ∗∗ which looks like a restatement of Proposition , when this is written as ( ); but Euclid derives the result afresh, using that, by definition, k 1: A :: C : A C, 1: A :: D : A D, × × so that C : A C :: D : A D (since sameness of ratio is immediately transitive), and× therefore, alternately,× ( ). By applying Proposition  to this, we obtain Proposition , ∗∗

A : B :: A C : B C. (††) × × From ( ) and (††), along with the rule ∗∗ ′ ′ E : F :: E : F = F = F , ⇒ we obtain A : B :: C : D A D = B C, ( ) ⇐⇒ × × ‡‡ which is Proposition . Euclid’s argument is as we have put it, although, when we writes out Propositions  and , he does not do as we have done, choosing the letters in ( ) and (††) so that Proposition  follows almost visually as ( ). ∗∗ In Proposition ,‡‡ we suppose C and D are the least X and Y such that X : Y :: A : B. If C is parts of A, then for some part E of A and F of B, we have C = kE and D = kF ; but then by Proposition ,

E : F :: C : D :: A : B, contrary to the minimality of C and D. Thus C is not parts, but is a part of A, and D is the same part of B; or as Euclid says, C measures A and D measures B equally (ἰσάκις). . Euclid’s Lemma 

There is no claim in Proposition  that numbers that are minimal as described must exist. Proposition  establishes a sufficient condition for their existence. If Α and Β are prime to one another, then they must be the least Ξ and Υ such that Ξ : Υ :: Α : Β. For, suppose not, but Γ and ∆ are least.

ἰσάκις ἄρα ὁ Γ τὸν Α μετρεῖ καὶ ὁ ∆ τὸν Β. ὁσάκις δὴ ὁ Γ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε. καὶ ὁ ∆ ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας. καὶ ἐπεὶ ὁ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, καί ὁ Ε ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας. // Therefore Γ measures Α and ∆ measures Β equally. Then as many times as Γ measures Α, so many units let there be in Ε. Therefore also ∆ measures Β according to the units in Ε. And since Γ measures Α according to the units in Ε, also therefore Ε measures Α according to the units in Γ.

This is not the language of the enunciation of Proposition , but is the language of its demonstration. At present, for some Ε we have Ε Γ = Α, and therefore Γ Ε = Α, that is, Ε measures Α; and similarly the× same Ε measures Β, which× is absurd. We jump ahead to Proposition , which is immediate from the defi- nitions:

῞Απας πρῶτος ἀριθμὸς πρὸς ἅπαντα ἀριθμόν, ὃν μὴ μετρεῖ, πρῶτός ἐστιν. // Every prime number, to every number that it does not measure, is prime.

Finally then, in Proposition , we suppose P is prime and measures A B, but does not measure A, so that P is prime to A by Proposition .× By hypothesis, for some C, C P = A B, so by Proposition , × × P : A :: B : C.

By Proposition , P and A are minimal; then by Proposition , P measures B.   Euclid’s Foundations of Arithmetic

. Symbolic mathematics

The last three propositions of Book vii of the Elements are hints of a symbolic mathematics, a mathematics of manipulating symbols that do not, or do not yet, have meaning. Such mathematics might be called an- alytic, as Descartes’s geometry is called; but it certainly did not originate with Descartes. Pappus describes it [, pp. –]: ἐν μὲν γὰρ τῇ ἀναλύσει τὸ ζητούμενον ὡς γεγονὸς ὑποθέμενοι τὸ ἐξ οὗ τοῦτο συμβαίνει σκοπούμεθα. // For in analysis we suppose that which is sought to be already done, and we inquire what it is from which this comes about. Pappus attributes the work of analysis that he describes to “Euclid the writer of the Elements, Apollonius of Perga and Aristaeus the elder.” Proposition vii. of the Elements has the enunciation, ᾿Εὰν ἀριθμος ὑπό τινος ἀριθμοῦ μετρῆται, ὁ μετρούμενος ὁμώνυμον μέρος ἕξει τῷ μετροῦντι. // If a number be measured by some number, the measured number has a part having the same name with the measuring number. Instead of having the same name, we could use “homonymous” for Eu- clid’s ὁμώνυμος. If A is measured by the number three, that is, by a triad B, then A = C B for some number C; but then also, by commutativity of multiplication,× A = B C, or A = 3C, so C is the third part of A. Here the triad and the third× are said to have the same name, or to be homonymous. Proposition vii. is the converse of : ᾿Εὰν ἀριθμος μέρος ἔχῃ ὁτιοῦν, ὑπὸ ὁμωνύμου ἀριθμοῦ μετρηθήσεται τῷ μέρει. // If a number have any part whatever, it will be measured by a [number] having the same name with the part. Proposition vii. is then a complement to , which is to find the least common multiple of given numbers: so  is to find the least number having given parts. The argument is thus: ῎Εστω τὰ δοθέντα μέρη τὰ Α, Β, Γ· δεῖ δὴ ἀριθμὸν εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ Α, Β, Γ μέρη. ῎Εστωσαν γὰρ τοῖς Α, Β, Γ μέρεσιν ὁμώνυμοι . Symbolic mathematics 

ἀριθμοὶ οἱ ∆, Ε, Ζ, καὶ εἰλήφθω ὑπὸ τῶν ∆, Ε, Ζ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Η. ῾Ο Η ἄρα ὁμώνυμα μέρη ἔχει τοῖς ∆, Ε, Ζ. τοῖς δὲ ∆, Ε, Ζ ὁμώνυμα μέρη ἐστὶ τὰ Α, Β, Γ· ὁ Η ἄρα ἔχει τὰ Α, Β, Γ μέρη. // Let the given parts be Α, Β, and Γ. We must find a number that will minimally have parts Α, Β, and Γ. So let the numbers having the same names as Α, Β, and Γ be ∆, Ε, and Ζ; and let the least number measured by ∆, Ε, and Ζ have been taken, namely Η. Then Η has parts having the same names as ∆, Ε, and Ζ. But the parts having the same names as ∆, Ε, and Ζ are Α, Β, and Γ. Therefore Η has the parts Α, Β, and Γ.

Moreover, Η must be the least number having these parts; for, if Θ has these parts and is less, then it also is measured by ∆, Ε, and Ζ, which is absurd. Such is the argument of the last proposition of Book vii. The diagram for this proposition consists of eight separate line seg- ments, labelled Α through Θ. What can the first three of these be—the ones that are called the given parts? At the beginning, they are not known to be parts of anything; they are just parts, simply, like a half or a third or a fourth. Should they be thought of as segments having frac- tional length? Are they ratios as such, rendered concrete for the moment so that they can be talked about, but not intended for comparison with particular numbers? In the latter case, this would be the only instance I know of where a ratio can actually be pointed to in a diagram; elsewhere, a ratio is a relation between two things in a diagram. Thus I have to wonder whether Proposition  is not a later addition to Book vii, by somebody thinking there should be a formal correlate to Proposition . Compare the exposition of Proposition :

᾿Αριθμὸς γὰρ ὁ Α μέρος ἐχέτω ὁτιοῦν τὸν Β, καὶ τῷ Β μέρει ὁμώνυμος ἔστω [ἀριθμὸς] ὁ Γ. // For let Α have whatsoever part, Β; and let the number having the same name as Β be Γ.

Here Β is not a ratio; it is a part of a given number, namely Α. At the beginning of Proposition , again, there is nothing for Α, Β, and Γ to be part of: they are just “parts,” or rather each of them is just a “part.” What then can it mean to speak of numbers ∆, Ε, and Ζ having the same name with Α, Β, and Γ?   Euclid’s Foundations of Arithmetic

One might understand Α, Β, and Γ as divisors, so that the point of Proposition vii. is to find the least number divided by Α, Β, and Γ. But in that case, ∆, Ε, and Ζ would be respectively equal to Α, Β, and Γ; and they are not said to be equal. Neither do they appear equal in Heiberg’s diagram, though we do not know how well Heiberg’s diagrams conform with those of the manuscripts. The position of Proposition  at the end of Book vii may contribute to the plausibility of its being a later addition to the Elements. Per- haps  and  are also later additions. All three propositions follow easily by means of the commutativity of multiplication, and this com- mutativity is relied on implicitly throughout the ensuing Books viii and ix. Again, Proposition vii. in particular seems to be due to somebody who is thinking “formally”: since measuring is correlated with dividing by means of Proposition vii., for every proposition like  about measur- ing, a proposition like  about dividing can be written. Similar additions to Wikipedia are inserted today by people who want to make their con- tribution: these contributions do not always reflect a full understanding of the surrounding text. Bibliography

[] Archimedes. The Two Books On the Sphere and the Cylinder, vol- ume I of The Works of Archimedes. Cambridge University Press, Cambridge, . Translated into English, together with Euto- cius’ commentaries, with commentary, and critical edition of the diagrams, by Reviel Netz.

[] Aristotle. Categories, On Interpretation, and Prior Analytics. Num- ber  in Loeb Classical Library. Harvard University Press and William Heinemann Ltd, Cambridge, Massachusetts and London, . With English translation by H. P. Cooke and H. Tredennick.

[] Jeremy Avigad, Edward Dean, and John Mumma. A formal system for Euclid’s Elements. Rev. Symb. Log., ():–, .

[] Paul Bahn. Archaeology: A very short introduction. Very Short Introductions. Oxford University Press, Oxford, . Fully updated new edition. First edition, .

[] Andrew Barker, editor. Greek Musical Writings. Volume II. Har- monic and Acoustic Theory. Cambridge University Press, . First paperback edition .

[] Andrew Barker. The Science of Harmonics in Classical Greece. Cambridge University Press, .

[] Mary Beard. No more scissors and paste. London Review of Books,  March . Review of History Man: The Life of R.G. Colling- wood by Fred Inglis.

  Bibliography

[] Joseph Bertrand. Traité d’Arithmétique. Hachette, Paris, . Electronic version from Gallica Bibiotèque Numérique (http:// gallica.bnf.fr).

[] Simon Blackburn. Collingwood, Robin George (–). In Rout- ledge Encyclopedia of Philosophy. Routledge, London and New York, . Version ..

[] Simon Blackburn. Being and time. The New Republic,  April . Review of History Man: The Life of R.G. Collingwood by Fred In- glis. Available from http://lclane2.net/collingwood.html, ac- cessed October , .

[] Roberta Botto Mura and Akbar Rhemtulla. Orderable Groups. Mar- cel Dekker, Inc., New York-Basel, . Lecture Notes in Pure and Applied Mathematics, Vol. .

[] David M. Burton. Elementary Number Theory. McGraw-Hill, Boston, sixth edition, .

[] Pierre Chantraine. Dictionnaire étymologique de la langue grecque. Histoire des mots. Klincksieck, Paris, –. In four volumes.

[] R. G. Collingwood. Speculum Mentis or The Map of Knowledge. Clarendon Press, Oxford, . Reprinted photographically in Great Britain at the University Press, Oxford, .

[] R. G. Collingwood. The Principles of Art. Oxford University Press, London, Oxford, and New York, paperback edition, . First published .

[] R. G. Collingwood. The Idea of Nature. Oxford University Press, London, Oxford, and New York, paperback edition, . First published .

[] R. G. Collingwood. An Autobiography. Clarendon Press, Oxford, . First published . With a new introduction by Stephen Toulmin. Reprinted . Bibliography 

[] R. G. Collingwood. The Idea of History. Oxford University Press, Oxford and New York, revised edition, . With Lectures – . Edited with an introduction by Jan van der Dussen.

[] R. G. Collingwood. An Essay on Metaphysics. Clarendon Press, Oxford, revised edition, . With an Introduction and additional material edited by Rex Martin. Published in paperback . First edition .

[] R. G. Collingwood. The Principles of History and other writings in philosophy of history. Oxford, . Edited and with an introduction by W. H. Dray and W. J. van der Dussen.

[] R. G. Collingwood. An Essay on Philosophical Method. Clarendon Press, Oxford, new edition, . With an Introduction and ad- ditional material edited by James Connelly and Giuseppina D’Oro. First edition .

[] Richard Dedekind. Essays on the Theory of Numbers. I: Continuity and Irrational Numbers. II: The Nature and Meaning of Numbers. authorized translation by Wooster Woodruff Beman. Dover Publi- cations Inc., New York, .

[] René Descartes. The Geometry of René Descartes. Dover Publica- tions, Inc., New York, . Translated from the French and Latin by David Eugene Smith and Marcia L. Latham, with a facsimile of the first edition of .

[] Jean Dubois, Henri Mitterand, and Albert Dauzat. Dictionnaire d’Étymologie. Larousse, Paris, . First edition .

[] Sextus Empiricus. Sextus Empiricus. Ge. Reimer, Berlin, . Ex recensione Immanuelis Bekkeri.

[] Sextus Empiricus. Against the Professors. Number  in Loeb Clas- sical Library. Harvard University Press, Cambridge, Massachusetts, and London, England, . With English translation by R. G. Bury.  Bibliography

[] Euclid. Euclidis Elementa, volume I of Euclidis Opera Omnia. Teub- ner, Leipzig, . Edited with Latin interpretation by I. L. Heiberg. Books I–IV.

[] Euclid. The Thirteen Books of Euclid’s Elements. Dover Publica- tions, New York, . Translated from the text of Heiberg with introduction and commentary by Thomas L. Heath. Republication of the second edition of .

[] Euclid. Στοιχεία Εὐκλείδου. http://users.ntua.gr/dimour/ euclid/, . Edited by Dimitrios E. Mourmouras. Accessed De- cember , .

[] Euclid. Öklid’in Öğelerinin  Kitabından Birinci Kitap. Math- ematics Department, Mimar Sinan Fine Arts University, Istanbul, th edition, September . The first of the  books of Euclid’s Elements. Greek text, with Turkish version by Özer Öztürk & David Pierce.

[] David Fowler. The Mathematics of Plato’s Academy. Clarendon Press, Oxford, second edition, . A new reconstruction.

[] H. W. Fowler. A Dictionary of Modern English Usage. Oxford University Press, London, . Corrected reprint of .

[] Carl Friedrich Gauss. Disquisitiones Arithmeticae. Carl Friedrich Gauss Werke. Gerh. Fleischer Jun., Leipzig, . Electronic ver- sion of the original Latin text from Goettingen State and University Library.

[] Carl Friedrich Gauss. Disquisitiones Arithmeticae. Springer-Verlag, New York, . Translated into English by Arthur A. Clarke, re- vised by William C. Waterhouse.

[] Timothy Gowers. Mathematics: A very short introduction. Very Short Introductions. Oxford University Press, Oxford, . Bibliography 

[] Reha Günay. A Guide to the Works of Sinan the Aarchitect in Is- tanbul. YEM, Istanbul, July . [] G. H. Hardy. A Mathematician’s Apology. Cambridge University Press, Cambridge, . With a foreword by C. P. Snow. Reprint of the  edition. First edition . [] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon Press, Oxford, fifth edition, . First edition . Reprinted . [] Thomas Heath. A History of Greek mathematics. Vol. I. Dover Publications Inc., New York, . From Thales to Euclid. Corrected reprint of the  original. [] Richard D. Heffner. A Documentary History of the United States. New American Library, New York, rd edition, . Expanded and Revised Bicentennial Edition. [] Hero. Heronis Alexandrini opera quae supersunt omnia. Volumen IV. Heronis definitiones cum variis collectionibus. Heronis quae feruntur geometrica. Teubner, Stuttgart, . Copiis Guilelmi Schmidt usus edidit J. L. Heiberg. Facsimile reproduction of  edition. Electronic version created at Gallica, downloaded from wilbourhall.org. [] David Hilbert. The Foundations of Geometry. Authorized transla- tion by E. J. Townsend. Reprint edition. The Open Court Publishing Co., La Salle, Ill., . Based on lectures –. Translation copy- righted . Project Gutenberg edition released December ,  (www.gutenberg.net). [] T. F. Hoad, editor. The Concise Oxford Dictionary of English Ety- mology. Oxford University Press, Oxford, . [] Iamblichus. In Nicomachi Arithmeticam Introductionem Liber. Teubner, Leipzig, . Ad fidem Codicis Florentini edidit Hermenegildus Pistelli.  Bibliography

[] Thomas J. Jech. The Axiom of Choice. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, . Studies in Logic and the Foundations of Mathe- matics, Vol. .

[] J. P. Kenyon, editor. Dictionary of British History. Wordsworth, Ware, Hertfordshire, . First published  by Market House Books.

[] T. Y. Lam. A First Course in Noncommutative Rings, volume  of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, .

[] Edmund Landau. Elementary Number Theory. Chelsea Publishing Co., New York, N.Y., . Originally part of Vorlesungen über Zahlentheorie (Leipzig, ). Translated by J. E. Goodman.

[] Edmund Landau. Foundations of Analysis. The Arithmetic of Whole, Rational, Irrational and Complex Numbers. Chelsea Pub- lishing Company, New York, N.Y., third edition, . Translated by F. Steinhardt; first edition ; first German publication, .

[] Serge Lang. Algebra. Addison-Wesley, Reading, Massachusetts, third edition, . Reprinted with corrections, .

[] William J. LeVeque. Elementary Theory of Numbers. Dover, New York, . Original published by Addison-Wesley in . Slightly corrected.

[] Henry George Liddell and Robert Scott. A Greek-English Lexicon. Clarendon Press, Oxford, . Revised and augmented throughout by Sir Henry Stuart Jones, with the assistance of Roderick McKenzie and with the cooperation of many scholars. With a revised supple- ment.

[] Barry Mazur. How did Theaetetus prove his theorem? In P. Kalka- vage and E. Salem, editors, The Envisioned Life: Essays in honor Bibliography 

of Eva Brann. Paul Dry Books, . http://www.math.harvard. edu/~mazur/preprints/Eva.pdf, accessed September , . [] Barry Mazur. When is one thing equal to some other thing? In Proof and other dilemmas, MAA Spectrum, pages –. Math. Assoc. America, Washington, DC, . [] James Morwood and John Taylor, editors. Pocket Oxford Classical Greek Dictionary. Oxford University Press, Oxford, . [] David Mumford. Trends in the profession of mathematics. Mitt. Dtsch. Math.-Ver., ():–, . [] Murray et al., editors. The Compact Edition of the Oxford English Dictionary. Oxford University Press, . [] Reviel Netz. The Shaping of Deduction in Greek Mathematics, vol- ume  of Ideas in Context. Cambridge University Press, Cambridge, . A study in cognitive history. [] Nicomachus of Gerasa. Introduction to Arithmetic, volume XVI of University of Michigan Studies, Humanistic Series. University of Michigan Press, Ann Arbor, . Translated by Martin Luther D’Ooge. With studies in Greek arithmetic by Frank Egleston Rob- bins and Louis Charles Karpinski. First printing, . [] David Pengelley and Fred Richman. Did Euclid need the Euclidean algorithm to prove unique factorization? Amer. Math. Monthly, ():–, . [] David Pierce. Model-theory of vector-spaces over unspecified fields. Arch. Math. Logic, ():–, . [] David Pierce. Induction and recursion. The De Morgan Jour- nal, ():–, . http://education.lms.ac.uk/2012/04/ david-pierce-induction-and-recursion/. [] David Pierce. Abscissas and ordinates. Journal of Humanistic Math- ematics, x.  Bibliography

[] Robert M. Pirsig. Lila. Bantam, New York, .

[] Plato. Euthyphro, Apology, Crito, Phaedo, Phaedrus. Number  in Loeb Classical Library. Harvard University Press and William Heine- mann Ltd., Cambridge, Massachusetts, and London, . with an English Translation by Harold North Fowler, first printed .

[] Proclus. A Commentary on the First Book of Euclid’s Elements. Princeton Paperbacks. Princeton University Press, Princeton, NJ, . Translated from the Greek and with an introduction and notes by Glenn R. Morrow. Reprint of the  edition. With a foreword by Ian Mueller.

[] Procopius. On Buildings. Number  in Loeb Classical Library. Harvard University Press, Cambridge, Massachusetts, and London, England, . With an English translation by H. B. Dewing. Reprinted with revisions .

[] Lucio Russo. The Forgotten Revolution. Springer-Verlag, Berlin, . How science was born in  BC and why it had to be reborn. Translated from the  Italian original by Silvio Levy.

[] Walter W. Skeat. A Concise Etymological Dictionary of the English Language. Perigee Books, New York, . Original date of this edition not given. First edition .

[] Herbert Weir Smyth. Greek Grammar. Harvard University Press, Cambridge, Massachussets, . Revised by Gordon M. Messing, . Eleventh Printing. Original edition, .

[] Ivor Thomas, editor. Selections Illustrating the History of Greek Mathematics. Vol. I. From Thales to Euclid. Number  in Loeb Classical Library. Harvard University Press, Cambridge, Mass., . With an English translation by the editor.

[] Ivor Thomas, editor. Selections Illustrating the History of Greek Mathematics. Vol. II. From Aristarchus to Pappus. Number  in Bibliography 

Loeb Classical Library. Harvard University Press, Cambridge, Mass, . With an English translation by the editor.

[] Gore Vidal. Julian. Abacus, London, . Reprinted .

[] Karl von Jan, editor. Musici Scriptores Graeci. Aristoteles, Eu- clides, Nicomachus, Bacchius, Gaudentius, Alypius. Et Melodiarum Veterum Quidquid Exstat. Teubner, Leipzig, .

[] William Wordsworth. Intimations of Immortality From Recollections of Early Childhood and Other Poems. Houghton Mifflin, Boston, .

[] E. C. Zeeman. What’s wrong with Euclid book V. Bull. London Math. Soc., :–, . The  David Crighton Lecture.